Alterations and resolution of singularities

Dan Abramovich and Frans Oort

June 4, 1998

Contents

0 Introduction 4 0.1 The alteration paradigm . . . . . . . . . . . . . . . . . . . . . . . 4 0.2 The purpose of this paper . . . . . . . . . . . . . . . . . . . . . . 4 0.3 Historical context . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 0.4 Comparison of approaches . . . . . . . . . . . . . . . . . . . . . . 5 0.5 A sketch of the construction of an alteration giving a regular variety. 6

I The alteration theorem 10

1 Some preliminaries and generalities on varieties 10 1.1 Varieties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2 Operations on varieties. . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Smooth morphisms and regular varieties . . . . . . . . . . . . . . 11 1.4 Resolution, weak and strong. . . . . . . . . . . . . . . . . . . . . 11 1.5 Normal crossings. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6 Flatness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.7 Stable curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.8 Minimal models, existence and uniqueness. . . . . . . . . . . . . 13

2 Results 13

3 Some tools 15 3.1 The strict transform. . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Chow’s lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 The flattening lemma. . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Deforming a node . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5 Serre’s lemma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.6 Extending stable curves. . . . . . . . . . . . . . . . . . . . . . . . 19 3.7 Contraction and stabilization. . . . . . . . . . . . . . . . . . . . . 20

4 Proof of de Jong’s main theorem 21 4.1 Preparatory steps and observations . . . . . . . . . . . . . . . . . 21 4.2 Producing a projection . . . . . . . . . . . . . . . . . . . . . . . . 22 4.3 Enlarging the divisor Z . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 The idea of simplifying the fibers . . . . . . . . . . . . . . . . . . 24 4.5 Straightenning out Z. . . . . . . . . . . . . . . . . . . . . . . . . 25 4.6 Producing a family of stable pointed curves. . . . . . . . . . . . . 25 4.7 The three point lemma. . . . . . . . . . . . . . . . . . . . . . . . 26 4.8 Flattening the graph. . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.9 Using the three point assumption . . . . . . . . . . . . . . . . . . 27 4.10 Induction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.11 Exercises on simple blowup of nodal families. . . . . . . . . . . . 27

5 Modifications of the proof for Theorems 2.5 and 2.8. 29 5.1 Exercises on removing the conditions on the projection. . . . . . 29 5.2 Exercises on Theorem 2.5. . . . . . . . . . . . . . . . . . . . . . . 29 5.3 Exercises on theorem 2.8 . . . . . . . . . . . . . . . . . . . . . . . 30

6 Toroidal geometry 30 6.1 Basic definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.2 The cone. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.3 The toroidal picture . . . . . . . . . . . . . . . . . . . . . . . . . 32 6.4 Birational affine morphisms. . . . . . . . . . . . . . . . . . . . . . 32 6.5 Principal affine opens. . . . . . . . . . . . . . . . . . . . . . . . . 33 6.6 Fans and polyhedral complexes. . . . . . . . . . . . . . . . . . . . 33 6.7 Modifications and subdivisions. . . . . . . . . . . . . . . . . . . . 33 6.8 Nonsingularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.9 Desingularization . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 6.10 Exercises on toric varieties and toroidal embeddings. . . . . . . . 34 6.11 Abhyankar’s lemma in toroidal terms. . . . . . . . . . . . . . . . 35

7 Weak resolution of singularities I 35 7.1 Projection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 7.2 Vector bundles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7.3 Conclusion of the proof. . . . . . . . . . . . . . . . . . . . . . . . 36

8 Weak resolution of singularities II 37 8.1 Reduction steps. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 8.2 The ideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

9 Intersection multiplicities 38

II Moduli of curves 40

10 Introduction to moduli of curves 40 10.1 The functor of points and representability . . . . . . . . . . . . . 40 10.2 Moduli functors and fine moduli schemes. . . . . . . . . . . . . . 41 10.3 Historical interlude. . . . . . . . . . . . . . . . . . . . . . . . . . 42 10.4 Coarse moduli schemes . . . . . . . . . . . . . . . . . . . . . . . . 42

11 Stable reduction and completeness of moduli spaces 45 11.1 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 11.2 Stable reduction for elliptic curves. . . . . . . . . . . . . . . . . . 48 11.3 Remarks about monodromy . . . . . . . . . . . . . . . . . . . . . 50

12 Construction of moduli spaces. 51 12.1 Geometric Invariant Theory and Gieseker’s approach. . . . . . . 52 12.2 Existence of \overline{M_{g,n}}. . . . . . . . . . . . . . . . . . . . . . . . . . . 53

13 Existence of tautological families. 53 13.1 Hilbert schemes and level structures. . . . . . . . . . . . . . . . . 53 13.2 Moduli with level structure. . . . . . . . . . . . . . . . . . . . . . 54 13.3 Proof by reduction to the unpointed case . . . . . . . . . . . . . 55 13.4 Artin’s approach via slicing . . . . . . . . . . . . . . . . . . . . . 56

14 Moduli, Automorphisms, and families 56

0 Introduction

H. Hironaka, 1964:

In characteristic zero any variety can be modified into a nonsingular variety.

A. J. de Jong, 1995:

Any variety can be altered into a nonsingular variety.

On July 26, 1995, at the University of California, Santa Cruz, a young Dutch mathematician by the name Aise Johan de Jong made a revolution in the study of the arithmetic, geometry and cohomology theory of varieties in positive or mixed characteristic. The talk he delivered, first in a series of three entitled “Dominating Varieties by Smooth Varieties”, had a central theme: a systematic application of fibrations by nodal curves. Among the hundreds of awe struck members of the audience, participants of the American Mathematical Society Summer Research Institute on Algebraic Geometry, many recognized the great potential of Johan de Jong’s ideas even for complex algebraic varieties, and indeed soon more results along these lines began to form.

0.1 The alteration paradigm

A.J. de Jong’s main result was, that for any variety X, there is a nonsingular variety Y and an alteration, namely a proper, surjetive and generically finite morphism, Y → X (see Theorem 2.4 for a precise statement). This is in contrast with Hironaka’s result, which uses only a modification, namely a proper birational morphism.

Here is the basic structure of the proof by De Jong:

• Projection. For a given variety X of dimension d we produce a morphism f : X → P with dim P = d − 1, and all fibres of f are curves (we may first have to apply a modification to X).

• Desingularization of fibers. After an alteration of the base P, we arrive at a new morphism f : X → P where all fibres are curves with only ordinary nodes as singularities. The main tool here is the theory of moduli of curves.

• Desingularization of base. After a further alteration on the base P, we arrive at a new morphism f : X → P as above, where P is regular. Here we use induction, i.e. supposing that the theorem is already true for varieties of dimension d − 1. So here we “desingularize the base”.

• Desingularization of total space. Given the last two steps, an explicit and easy method of resolution of singularities finishes the job.

0.2 The purpose of this paper

This paper is an outgrowth of our course material prepared for the Working Week on Resolution of Singularities, which was held during September 7-14, 1997 in Obergurgl, Tirol, Austria. As we did in the workshop, we intend to explain Johan de Jong’s results in some detail, and give some other results following the same paradigm, as well as a few applications, both arithmetic and in characteristic zero. We hope that the reader will come to share some of the excitement we felt on that beautiful July day in Santa Cruz.

In the rest of this introduction we give an overview of the proof and the material involved. We hope that this introduction will give most readers a general feeling of what the results are about. The body of the paper is divided in two parts. We begin part I by expanding on some of the preliminary material necessary for understanding the proofs by any student of algebraic geometry. Then we go back to the proof of de Jong’s main theorem, as well as some generalizations. Proofs of some variants and generalizations of de Jong’s theorems are indicated in the form of exercises, with sufficient hints and references, which we hope will enable the reader to appreciate de Jong’s work. Part II is an introduction to an ingredient of the proof - the theory of moduli of curves. We aim to indicate the main ideas behind the proofs of the main theorems about existence and properties of moduli spaces, again accompanied with a collection of exercises.

As a result, this account is mostly expository. The only point where some novelty appears is in Section 13, where we show the existence of tautological families of stable curves over the moduli spaces of stable pointed curves with level structure. This has been “well known to the experts” for years, and can be collected from the litterature. However a complete account under one roof has not been published.

0.3 Historical context

There are many cases in geometry in which one wants to transform a singular variety into a non-singular one: once arrived in such a situation, various technical steps can be performed, not possible on singular varieties.

Since the beginning of the century, partial results in this direction appeared, crowned by Hironaka’s theorem on resolution of singularities in characteristic zero, in 1964.

Hironaka’s ingenuous proof had many applications, but it was not easy to understand the fine details of his proof. Generalizing that method to varieties in positive characteristic has failed up to now. Indeed, resolution of singularities in positive characteristic has been a topic to which many years of intensive research have been devoted, and up to now the status is not yet clear: for the general question of resolution of singularities in positive characteristic we have neither a fully verified theorem nor a counterexample. In addition, the algorithms involved in Hironaka’s theory were difficult to generalize, even in characteristic 0, to some important more complicated situations.

It seemed that a lull in development of this subject had been reached, until a totally new idea came about. In 1995 Johan de Jong approached the problem above, of transforming a variety into a nonsingular one, from a different angle. The idea of the proof is surprisingly easy, and for many applications his result is sufficient. His approach is very geometric, and hence it works in a wide range of situations. The alteration paradigm automatically works in all characteristics, and a suitable version works in mixed characteristic as well. It easily gives rise to some new “semistable reduction” type results which are new even over the complex numbers. Moreover, without much effort it give birth to new, “conceptually easy” proofs of a weaker form of Hironaka’s theorem.

0.4 Comparison of approaches

Let us take a moment to make a qualitative comparison of Hironaka’s result and de Jong’s result.

In the approach taken by Hironaka, singularities of a variety are studied closely, invariants measuring the difficulty of the singularities are defined, and a somewhat explicit algorithm is applied in order to improve the singularities, in the sense that the given invariants get “better”. One needs to show that the algorithm terminates (and indeed in characteristic zero it does), resulting in the construction of a regular variety. A big advantage of this process developed by Hironaka (and by many others) is the fact that usually it is very explicit, it is canonical in a certain sense and once it works, the result is in its strongest form, see [68] and [11], as well as [19] in this volume.

In the approach by Johan de Jong, the singularities are, at first, completely ignored. The idea is to first bring the variety to a special form: a fibration by nodal curves. Here one pays a big price: in order to arrive at this special form one needs to use an operation - called alteration - which extends the function field of the variety. However, once we arrive at this form, we can use induction on the dimension for the base space of the fibration, and automatically arrive at a situation where the variety has very mild singularities. Only then, finally, attention is paid to the singularities. But these are so mild that an easy and explicit blowing up finishes the job.

0.5 A sketch of the construction of an alteration giving a regular variety.

Here we give a much simplified form of the proof of A.J. de Jong’s main Theorem (Theorem 2.4 in this text). We break up the proof in steps. A star attached to a step means that in that phase of the proof a finite extension of the function field might be involved, i.e. the alteration constructed might not be a modification. In steps without a star only modifications are used.

Before starting, a small technical point is necessary. In the course of the proof we use induction on the dimension of the variety X, and it turns out that for the induction to work we need the statement of the theorem to involve a closed subset Z ⊂ X as well. Our final goal will be to find an alteration f : Y → X such that f^{−1}Z is a normal crossings divisor.

We start with a field k, a variety X and a closed subset Z ⊂ X, over the field k.

Step 0. We can reduce to the case where k is algebraically closed, the variaty X is projective and normal, and the closed subset Z a divisor.

We intend to say: if we prove the theorem with this new additional data, then the theorem in the original, more general form follows. Reducing to the algebraically closed field case is standard - in the main body of the paper we avoid it, assuming k is algebraically closed. The main ingredient for projectivity is Chow’s Lemma (see [Red Book], pp. 85-89, or [HAG], Exercise II.4.10): for a variety X over K, there exists a modification X′ → X, such that X′ is quasiprojective. To make Z into a divisor we simply blow it up inside X. Replacement convention. From now on, in each step, we shall replace X by a new variety X′ over k which admits a modification or an alteration X′ → X, arriving finally at a regular variety and an alteration of the variety produced in Step 0.

Step 1. After modifying X, construct a morphism f : X → P of projective varieties whose generic fiber is an irreducible, complete, non-singular curve.

Note: dim(P) = dim(X) − 1, which suggests using induction later.

Actually we need a little more, but the technical details will be discussed in the main text.

This step follows a classical, geometric idea. Set dim(X) = d, and assume X ⊂ ℙ^{N}. Using Bertini’s theorem we see that we can find a linear subvariety L ⊂ ℙ^{N} “in general position” with dim(L) = N −d such that the projection with center L gives a rational map X −−→ ℙ^{d−1} where the generic fibre is a regular curve. After modifying X we can make this rational map into a morphism. The strict transform. We will use an operation which de Jong called the “strict transform”. (In [10], 815-12 the terminology “strict alteration” is used). Consider a morphism X → S, and a base change T → S. Assume T to be integral, and let η ∈ T be its generic point. Then define X′ ⊂ T ×_{S} X as the closure of the generic fibre (T ×_{S} X)_{η} in (T ×_{S} X). A more thorough discussion of this operation will follow in Section 3.1.

In our situation X → P, we will often replace P by an alteration, and then simply replace X by its strict transform.

Step 2^{∗}. After applying alterations to X and to P we can arrive at a morphism f : X → P as in Step 1, and disjoint sections σ_{1}, ⋯, σ_{n} : P → X, such that every geometric component C′ of every geometric fibre of f meets at least three of these sections in the smooth locus of f, i.e. in C′ ∩ Sm(f).

There is a “multi-section” in the situation of Step 1 having this property. After an alteration on Y and on X this becomes a union of sections. Stable pointed curves. An algebraic curve is called nodal if it is complete, connected and if the singularities of C are not worse than ordinary double points. Its arithmetic genus is given by g = dim_{k} H^{1}(C, O_{C}).

Suppose C is a nodal curve of genus g over a field k, and let P_{1}, ⋯, P_{n} ∈ C(k) with 2g − 2 + n > 0; we write P = {P_{1}, ⋯, P_{n}}; this is called a stable n-pointed curve if:

• the points are mutually different, i < j ⟹ P_{i} ≠ P_{j},

• none of these marked points is singular, P_{i} ∉ Sing(C),

• and Aut(C, P) is a finite group; under the previous conditions (and k algebraically closed) this amounts to the condition that for every regular rational irreducible component

ℙ_{1} ≅ C′ ⊂ C, then #(C′ ∩ (P ∪ Sing(C))) ≥ 3.

A flat family of curves is called “a family of stable n-pointed curves” if all geometric fibres are stable n-pointed curves in the sense just defined, the markings given by sections.

Historically, stable curves and stable pointed curves were introduced in order to construct, in a natural way, compactifications of moduli spaces (see [17]). Certainly the following names should be mentioned: Zariski, A. Mayer, Deligne, Mumford, Grothendieck, Knudsen, and many more. It came a bit as a surprise when de Jong used these for a desingularization-type problem!

Step 3^{∗}. After an alteration on the base P, we can assume that X → P is a projective family of stable n-pointed curves.

We briefly sketch the heart of the proof of this step - it will be discussd in detail later. Extending families of curves. We need the following fundamental fact: suppose we are given a variety P, an open dense subset U ⊂ P, and a family of stable curves C_{U} → U:

\begin{aligned}C_{U} & ⊂ & ? \\ ↓ & & ↓ \\ U & ⊂ & P. \end{aligned}

Then there is an alteration a : P_{1} → P such that the pullback family C_{U_{1}} → U_{1} over the open set U_{1} = a^{−1}U can be extended to a family of stable curves C_{1} → P_{1}:

\begin{aligned}C_{U_{1}} & ⊂ & C_{1} \\ ↓ & & ↓ \\ U_{1} & ⊂ & P_{1}. \end{aligned}

The first result behind this is the existence of a moduli space of stable curves ([39], see also Section 12). Then one needs the fact that a finite cover M → \overline{M_{g,n}} of the moduli space admits a “tautological family” - namely, a family C → M such that the associated morphism M → \overline{M_{g,n}} is the given finite cover. One could consult [16] (the precise statement we need follows from that paper), or use [21], where a tautological family of nodal curves is constructed over a moduli space of stable curves with a level structure.

The markings of the family X′ → P correspond to those of the stable n-pointed curve C → P, under the birational transformation thus defined. We want to show this extends to a morphism C → X′. Flattening of the graph. We take the closure T ⊂ X ×_{P} C of the graph of β_{0} : C_{U} → X_{U}, and apply the ”Flattening Lemma”, see 3.4 below. We arrive at new X, T, and C flat over P. All we have to show (modulo some technicalities) is that no point of a fibre of C → P is blown up to a component of a fibre of X → P. The Three Point Lemma. Using the markings, and studying carefully the geometry we show that indeed β_{0} extends to a morphism β. The crucial point here was that every component of every fiber of X over P has at least three nonsingular points marked by the sections σ_{i} (see 4.18 - 4.20 of [Alteration]).

Step 4^{∗}. After an alteration of P, we may assume that P is nonsingular.

We simply apply induction on the dimension of the base: we suppose that the theorem we want to prove is valid for all varieties having dimension less than dim X. Thus after an alteration of the base P we can suppose P is regular and the strict transform of X has all the previous properties. Following Z. The argument for the previous two steps should be carried through with a proper care given to the divisor Z. At the end, we can guarantee that Z is contained in the union of two types of sets:

• the images of the sections σ_{i}, and

• the inverse image of a normal crossings divisor Δ ⊂ P.

Moreover, in the induction hypothesis we can guarantee that the final family of curves X → P degenerates only over the normal crossings divisor δ.

Step 5. The singularities of the resulting family X → P are so mild that it is very easy to resolve them explicitly.

Indeed, each singular point can be described in formal coordinates be the equation xy = t_{1}^{k_{1}} ⋯ t_{r}^{k_{r}}. It is a fairly straightforward exercise to resolve these singularities.

Part I

The alteration theorem

1 Some preliminaries and generalities on varieties

1.1 Varieties.

To fix notation, we use the following definition of a variety:

Definition 1.1. By a variety defined over k we mean a geometrically integral scheme of finite type over k. If k ⊂ k_{1} we write X_{k_{1}} for X ×_{Spec k} Spec k_{1}.

In more down to earth terms this means: an affine variety defined over k is given as a closed subvariety of an affine space 𝔸_{k}^{n} defined by an ideal I ⊂ k[T_{1}, ⋯, T_{n}] = k[T] such that k_{1}·I ⊂ k_{1}[T] is a prime ideal for every (equivalently, for some) algebraically closed field k_{1} containing k. In general, a variety then is defined by gluing a finite number of affine varieties in a separated way. See [Red Book], I.5, Definition 1 (p. 35) and I.6, Definition 2 (p. 52).

Remark 1.2. This definition differs slightly from that in [Alteration]. De Jong requires the algebraic scheme to be integral, and we require that the schemes stay integral after extending the field. For example for any finite field extension k ⊂ K, the scheme Spec(K) is called a k-variety by de Jong, but we only say it is a variety defined over k if k = K. For most geometric situations the differences will not be important.

1.2 Operations on varieties.

Definition 1.3. A morphism of varieties Y → X is called a modification if it is proper and birational.

A modification is the type of “surgery operation” usually associated with resolution of singularities. Johan de Jong introduced the following important variant:

Definition 1.4 (de Jong). A morphism of varieties Y → X is called an alteration if it is proper, surjective and generically finite. This notion of alteration will also be used for integral schemes.

See [Alteration], 2.20.

Remark 1.5. A modification is a birational alteration.

Exercise 1.6. Show that an alteration φ : Y → X can be factored as

Y ⟶^{π} Z ⟶^{f} X,

where π is a modification, and f is a finite morphism.

Exercise 1.7. Suppose moreover that a finite group G acts on Y by automorphisms such that the field of invariants K(Y)^{G} contains the function field K(X). Formulate other factorizations of φ.

Remark 1.8. Given a variety X and a nonzero coherent ideal sheaf I ⊂ O_{X}, the blowing up Bl_{I}X = P roj_{X}(⊕_{j≥0}I^{j}) → X gives naturally a modification, such that the inverse image of I becomes invertible. If Z ⊂ X is a subscheme with ideal sheaf I, the blowing up Bl_{Z}(X) of X along Z is defined to be the blowing up Bl_{I}X.

See [HAG], II.7, p. 163.

1.3 Smooth morphisms and regular varieties

The terminology “smooth” will only be used in a relative situation. Thus a morphism can be smooth. The terminology “regular”, or “non-singular”, will be used in the absolute sense. Thus a variety can be regular. This means that for every point P in the variety the local ring at P is a regular local ring. If a morphism X → Spec(K) is smooth, then X is regular. It is not recommended to use the terminology “a smooth variety”, which can be misleading and confusing.

1.4 Resolution, weak and strong.

We can now state what we mean by a resolution of singularities. There are two variants we will use:

Definition 1.9. Let X be a variety. A resolution of singularities in the weak sense is a modification Y → X such that Y is nonsingular.

Definition 1.10. Let X be a variety. A resolution of singularities in the strong sense is a modification Y → X, which is an isomorphism over the nonsingular locus X_{reg}, such that Y is nonsingular.

1.5 Normal crossings.

The following type of “nice subschemes” of a variety are quite useful in desingularization problems and applications:

Definition 1.11. Let X be a variety. A subscheme Z ⊂ X is called a strict normal crossings divisor if for each point x ∈ Z, there is a regular system of parameters y_{1}, …, y_{k} for x in X (in particular the point x ∈ X is supposed to be a regular point on X), such that Z is given on a Zariski neighborhood of x by the equation y_{1} ⋯ y_{l} = 0.

Suppose furthermore we have a finite group acting on Z and X equivariantly: G ⊂ Aut(Z ⊂ X). We say that Z is a G-strict normal crossings divisor if it has normal crossings, and for any irreducible component Z′ ⊂ Z, the orbit ∪_{g∈G} g(Z′) is normal.

See [Alteration], 7.1.

Strict normal crossings divisors have played an important role in resolution of singularities, and are essential in the proof of de Jong’s result.

1.6 Flatness.

A crucial idea for studying “families of schemes” is Serre’s notion of flatness (see [HAG], III.9).

Definition 1.12. Let A be a ring and M and A-module. Recall that M is said to be a flat A-module if the functor N ↦ M ⊗_{A} N is exact.

A morphism of schemes X → Y is flat if at any point x ∈ X, whose image is y ∈ Y, the local ring O_{X,x} is a flat O_{Y,y}-module.

There are many important examples of flat morphisms which we will discuss later. The reader is advised to consult [HAG] or [42] for a more detailed discussion. The general picture should be that in a proper flat morphism, many esential numerical invariants (e.g. dimension, degree...) are “constant” from fiber to fiber, so we should really think about it as a “family”.

Here are some instructive examples of morphisms which are not flat:

Example 1.13. (See [HAG], III 9.7.1.) Let Y be a curve with a node (say, the locus xy = 0 in the affine plane). Let X → Y be the normalization (in the specific example, the disjoint union of two lines mapping onto the locus xy = 0). Then f : X → Y is not flat. The idea one should have in mind is that since over a general point in Y we have one point in X, and over the node we have two points in X, this is not really a nice family - it jups in degree.

The same reasoning gives a more general example:

Example 1.14. Let f : X → Y be a modification. Then f is flat if and only if it is an isomorphism.

In particular, a nontrivial blowup is never flat.

1.7 Stable curves.

We give a formal definition of the fundamental notion introduced in the introduction:

Definition 1.15. An S scheme C → S is called a family of nodal curves over S if it is of finite presentation, proper and flat, and all geometric fibers are connected reduced curves with at most ordinary double points (locally xy = 0) as singularities.

Remark 1.16. The terminology a nodal curve over S can be used interchangeably with a family of nodal curves over S. Indeed, if C → S comes by way of an extension of a nodal curve C_{η} over the generic point η_{S} of S, it may be natural to call it a nodal curve over S.

Definition 1.17. The discriminant locus Δ ⊂ S is the closed subset over which C → S is not smooth.

Definition 1.18 (Deligne and Mumford). A family of nodal curves

f : C → S,

together with sections s_{i} : S → C, i = 1, …, n with image schemes S_{i} = s_{i}(S), is called a family of stable n-pointed curves of genus g if

1. The schemes S_{i} are mutually disjoint.

2. The schemes S_{i} are disjoint from the non-smooth locus Sing(f).

3. All the geometric fibers have arithmetic genus g.

4. The sheaf ω_{C/S}(∑ S_{i}) is f-ample.

In case n = 0 we simply call these stable curves (rather than stable 0pointed curves).

The definition is made so that a stable pointed curve has a finite automorphism group (relative over S). It agrees with that made (informally) in the introduction.

Remark 1.19. In the litterature one sometimes finds the terminology “n-poin-ted stable curve” instead of “stable n-pointed curve”. We try to stick to the latter, since it effectively conveys the idea that the curve with the points is stable. The other terminology might give the impression we are dealing with stable curve with some points on them. This would be a different notion in general!

1.8 Minimal models, existence and uniqueness.

An important stepping stone for understanding moduli of stable curves is the notion of minimal models of 1-parameter families of curves.

Let K be a field, and C a complete, geometrically irreducible algebraic curve smooth over K; suppose the genus of C is at least 2. Let v be a discrete valuation on K, and R ⊂ K its valuation ring. Pick a projective model C_{0} of C over R. Following Abhyankar (1963) we can resolve singularities in dimension 2, therefore we may assume C_{0} is nonsingular. Following Shafarevich (1966) and Lipman (1969) we have the notion of the minimal model of C over S := Spec(R) (see [17], page 87). We thus arrive at a family of curves C → Spec(R) which is a regular 2-dimensional scheme, and which is relatively minimal.

Remark 1.20. Here we use a special case of resolution of singularities, namely in the case of schemes of dimension 2.

2 Results

First recall Hironaka’s theorem:

Theorem 2.1 (Hironaka). Let X be a variety over a field k of characteristic 0. Then there exists a sequence of modifications

X_{n} → X_{n−1} → ⋯ → X_{1} → X_{0} = X,

where each X_{i} → X_{i−1} is a blowing up with nonsingular center, and the center lies over the singular locus Sing(X). In particular, X_{n} → X is a resolution of singularities in the strong sense.

See [29]. Hironaka’s theorem and its refinements will be discussed elsewhere in this volume.

Our main goal is to prove the following result, due to A. J. de Jong:

Theorem 2.2. Let X be a variety over an algebraically closed field. There is a separable alteration Y → X such that Y is quasi projective and regular.

Corollary 2.3. Let X/k be a variety. There is a finite extension k ⊂ k_{1} and a separable alteration Y → X_{k_{1}} such that Y is quasi projective and regular.

In order for the induction in the proof to work, de Jong’s theorem gives more:

Theorem 2.4 (de Jong). Let X be a variety over an algebraically closed field, Z ⊂ X a proper closed subset. There is a separable alteration f : Y → X, and an open immersion j : Y ⊂ Ȳ, such that Ȳ is projective and regular, and the subset j(f^{−1}Z) ∪ (Ȳ Y ) is a strict normal crossings divisor.

See [Alteration], 4.1. The proof of this result will be given in Section 4. De Jong’s theorem has a few important variants. First, a theorem of

semistable reduction up to alteration over a one dimensional base:

Theorem 2.5 (de Jong). Let R be a discrete valuation ring, with fraction field K and residue field k. Let X → Spec R be an integral scheme of finite type such that X_{K} is a variety. There exists a finite extension R ⊂ R_{1}, where R_{1} is a discrete valuation ring with residue field k_{1}, and an alteration Y → X_{R_{1}}, such that Y is nonsingular, and the special fiber Y_{k_{1}} is a reduced, strict normal crossings divisor.

See [Alteration], 6.5. The proof is detailed in Section 5.

This theorem is belongs to a class of theorems about “desingularization of morphisms”. A “dual” case, which can actually serve as a building block in proving the alteration type theorems, is the case where the base is arbitrary dimensional, and the fibers are curves. A proof can be found in [31].

Theorem 2.6 (de Jong). Let π : X → B be a proper surjective morphism of integral schemes, with dim X = dim B + 1. Let Z ⊂ X be a proper closed subset. There exists an alteration B_{1} → B, a modification X_{1} → X̃_{B_{1}} of the strict transform X̃_{B_{1}} (see Definition 3.1), sections s_{i} : B_{1} → X_{1}, and a proper closed subset Σ ⊂ B_{1} such that

1. π_{1} : X_{1} → B_{1} is a family of pointed nodal curves,

2. s_{i} are disjoint sections, landing in the smooth locus of π_{1}, and

3. the inverse image Z_{1} of Z in X_{1} is contained in the union of π_{1}^{−1}Σ (the “vertical part”) and s_{i}(B_{1}) (the “horizontal part”).

The reader who has solved the exercises in Section 5 will be able to complete the proof of this theorem.

From this de Jong deduced the following refinement of Theorem 2.5:

Theorem 2.7 (de Jong). Let π : X → B be a proper surjective morphism of integral schemes, dim X = dim B +r. Assume that B admits a proper morphism to an excellent two-dimensional scheme S. Then there are alterations B_{1} → B and X_{1} → X̃_{B_{1}}, a factorization X_{1} → X_{2} → ⋯ → X_{r} → X_{r+1} = B, and sucbschemes Σ_{i} = Σ_{i}^{hor} ∪ Σ_{i}^{ver}, such that

1. X_{i} are nosingular and Σ_{i} are normal crossings divisors, i = 1, …, r + 1;

2. π_{i} : X_{i} → X_{i+1} are families of nodal curves, smooth away from Σ_{i+1}, and

3. Σ_{i}^{hor} is the union of disjoint sections of π_{i}, lying in the smooth locus of π_{i}.

See [31]. Alternative proofs of different versions of this theorem were provided in [1] and [43].

Next, we consider a finite group action:

Theorem 2.8 (de Jong). Let X be a variety over an algebraically closed field, Z ⊂ X a proper closed subset, G ⊂ Aut(Z ⊂ X). There is an alteration f : Y → X, and a finite subgroup G_{1} ⊂ Aut Y, satisfying:

1. there is a surjection G_{1} → G such that f is G_{1} equivariant, and the field extension K(X)^{G} ⊂ K(Y)^{G_{1}} is purely inseparable;

2. Y is quasi projective and nonsingular; and

3. f^{−1}Z is a G-strict normal crossings divisor.

See [Alteration], 7.3. The proof is detailed in exercises in Section 5.

Note that, taking G = {id}, this implies:

Corollary 2.9. Let X be a variety over an algebraically closed field. There is a purely inseparable alteration Y → X where Y is a quotient of a nonsingular variety by the action of a finite group.

Remark 2.10. For generalizations which combine both Theorem 2.5 and Theorem 2.8, see [31].

In characteristic 0, any purely inseparable alteration is birational, and the quotient singularities can be improved:

Theorem 2.11 (See [2] and [12]). Let X be a variety over an algebraically closed field of characteristic 0. Then there is a projective resolution of singularities in the weak sense Y → X.

Remark 2.12. This is a rather weak version of Hironaka’s theorem. The point is, that new proofs, by Abramovich and de Jong [2], and by Bogomolov and Pantev [12], were given based on de Jong’s ideas. The proof by Bogomolov and Pantev is extremely simple, drawing only on toric geometry. Its proof is detailed in Section 7.

Question 2.13. Can we improve the methods and obtain a weak resolution of singularities in all characteristics? Or, at least weak resolution up to purely inseparable alterations?

The proof by Abramovich and de Jong, detailed in Section 8, lends itself to generalizations in the flavor of de Jong’s semistable reduction theorem, such as the following two results:

Theorem 2.14 (Abramovich - Karu). Let X → B be a dominant morphism of complex projective varieties. There exists a commutative diagram

U_{X′} ⊂ X′ → X
↓ ↓ ↓
U_{B′} ⊂ B′ → B

such that

1. X′ → X and B′ → B are modifications,

2. X′ and B′ are nonsingular,

3. U_{X′} ⊂ X′ and U_{B′} ⊂ B′ are toroidal embeddings, and the morphism X′ → B′ is a toroidal morphism (see definition in 6).

Theorem 2.15 (Abramovich - Karu). Let X → B be a dominant morphism of complex projective varieties. There exists a commutative diagram

U_{X} ⊂ X_{1} → X
↓ ↓ π_{1} ↓ π
U_{B} ⊂ B_{1} → B

where B_{1} → B is an alteration, X_{1} → X̃_{B_{1}} is a modification of the strict transform, U_{X} ⊂ X_{1} and U_{B} ⊂ B_{1} are toroidal, the morphism π_{1} : X_{1} → B_{1} is toroidal with π_{1}^{−1}U_{B} = U_{X}, the variety B_{1} is nonsingular and

1. the morphism π_{1} is equidimensional and

2. all fibers of π_{1} are reduced.

See [3] for details. A refinement is given in [33], and an application in [34].

3 Some tools

In this section we gather some basic tools which we are going to use. Some of these tools seem to be of vital importance in algebraic geometry, and it is instructive to see them functioning in the context of de Jong’s theorem. We have included some indications of proofs for the interested reader. For the proof of the alteration theorem only the following will be necessary: Section 3.1, Lemmas 3.3, 3.4 and 3.9, and Theorem 3.12.

3.1 The strict transform.

See [Alteration], 2.18.

As mentioned in the introduction, we need an operation called the “strict transform”. Let us recall the definition.

Definition 3.1. Consider a morphism X → S, and a base change T → S. Assume T to be integral, and let η ∈ T be its generic point. Then define the strict transform X̃_{T} ⊂ T ×_{S} X as the Zariski closure of the generic fiber T ×_{S} η:

X̃_{T} =^{def} \overline{T ×_{S} η}^{Zar} ⊂ T ×_{S} X ⟶ X
↘ ↓ ↓
T  ⟶ S.

Note that if the image of η is not in the image of X → S (i.e. if T ×_{S} X → T is not dominant), then the strict transform in the sense explained here is empty.

Remark 3.2. The notion given here is different from the usual notion of the “strict transform” of a subvariety under a modification (compare with [HAG], II.7, the definition after 7.15). For example consider a blowing up T → S of a surface S in a point P ∈ S, and let C ⊂ S be a curve in S passing through P. The “strict alteration” (or “strict transform” in the terminology above) of C under T → S is empty; the “strict transform” of C under T → S in the classical sense, as explained in [HAG], II.7, is a curve in T.

Some people have suggested the use of terminology “essential pullback of X along T → S”, which may have some merits. After all, X̃_{T} contains only the “part” of T ×_{S} X which dominates T, which is in some sense its essential part.

3.2 Chow’s lemma.

An algebraic curve and a regular algebraic surface are quasi-projective. However in higher dimension an “abstract variety” need not be quasi-projective. A beautiful example by Hironaka (of a variety of dimension three) is described in [HAG], Appendix B, Example (3.4.1). However in certain situations (such as the alteration method described below) we like to work with projective varieties.

Lemma 3.3. Given a variety X, there is a modification Y → X such that Y is quasi-projective.

See [Red Book] , I.10, p. 85, or [HAG], Exc. II.4.10 p. 107.

3.3 The flattening lemma.

In some situations we want to replace a morphism by a flat morphism. One can show this is possible after a modification of the base. The general situation is studied in [59]. We only need this in an easier, special situation, as follows:

Lemma 3.4 (The Flattening Lemma). Let X and Z be varieties over a perfect field K (more generally, integral schemes of finite presentation) and X → Z a projective, dominant morphism. There exists a modification f : Y → Z such that the strict transform f′ : X̃_{Y} → Y is flat.

The main ingredient in the proof is the existence and projectivity of the Hilbert scheme. Hilbert schemes were introduced and constructed by Grothendieck in [24], Exp. 221 (see [46] for simplified proofs, [18] for discussion). We will come back to them in Section 10. Their purpose is to parametrize all subschemes of a fixed projective space. Of course, the set of all subschemes of a projective space is rather large, so we cut it down into bounded pieces by fixing the Hilbert polynomial P (T). Grothendieck’s result may be summarized as follows:

Theorem 3.5. There is a projective scheme H_{ℙ^{N} ,P (T)} over Spec ℤ and a closed subscheme X_{ℙ^{N} ,P (T)} ⊂ ℙ^{N} × H_{ℙ^{N} ,P (T)} which is flat over H_{ℙ^{N} ,P (T)}, such that H_{ℙ^{N} ,P (T)} parametrizes subschemes of ℙ^{N} with Hilbert polynomial P (T), and X_{ℙ^{N} ,P (T)} → H_{ℙ^{N} ,P (T)} is a universal family, in the following sense:

Given a scheme T, let X ⊂ ℙ^{N} × T be a closed subscheme which is flat over T and such that the Hilbert polynomial of the fibers is P (T). Then there exists a unique morphism h : T → H_{ℙ^{N} ,P (T)}, such that

X = T ×_{H_{ℙ^{N} ,P (T)}} X_{ℙ^{N} ,P (T)}

Back to the proof of Lemma 3.4. Note that the generic fibre of f is reduced. By generic flatness, there exists a dense, open subset i : U ↪ Z such that

f_{U} : X_{U} := X|_{U} ⟶ U

is flat. Let P be Hilbert polynomial of the fibres of f_{U} (all fibres in a flat family over an irreducible base have the same Hilbert polynomial), and let X → H be the universal family over the Hilbert scheme associated to this polynomial. We have a cartesian commutative diagram:

\begin{matrix}X_{U} & ⟶ & X \\ ↓ & & ↓ \\ U & ⟶^{g} & H. \end{matrix}

Note that X → H is a flat morphism. We have a morphism i × g : U → Z × H. Define

Z′ := \overline{i × g(U)}^{Zar} ⊂ Z × H,

and let X′ → Z′ be the pull back:

X′ = Z′ ×_{H} X .

Note that the base change of a flat morphism is flat, hence X′ → Z′ is flat. It follows from [HAG] III.9.8 that X′ is the strict transform under Z′ → Z of X → Z. 3.4

Remark 3.6. We cited [HAG] III.9.8, which is in fact an important building block in the construction of Hilbert schemes.

Remark 3.7. We can delete the word “dominant” in the flattening lemma, and still prove the conclusion, but we do not gain much: if X → Z is not dominant, the identity on Z gives a strict transform (in the sense explained above) of X such that X′ = ∅, and flatness trivially follows.

Remark 3.8. In the proof above we note a general method, which will also be used in the question of extending curves below: suppose we study a certain property (e.g. flatness of a map). Suppose there is a “universal family” having this property (e.g., the Hilbert scheme). Suppose also that in a given family the property holds over a dense open subset U in the base. Then, after a modification, or an alteration of the base, depending on the situation, we can achieve that property by mapping U to the base of the universal family, taking the closure of the graph, and pulling back the universal family.

3.4 Deforming a node

An important fact underlying the role of stable curves, which is implicitly invoked in several places in this paper, is that a node uv = 0 can only deform in a certain way. To be precise:

Lemma 3.9. Let R be a complete local ring with maximal ideal m and algebraically closed residue field. Let S = Spec R and denote teh special point by s. Let X → S be the completion of a nodal curve at a closed point x on the fiber X_{s} over s, so X_{s} = Spec(R/m[ū, v̄]/(ūv̄))^{∧}. Then there is an element f ∈ R, and liftings u of ū and v of v̄, such that X ≃ Spec(R[u, v]/(uv − f))^{∧}.

One can prove this using the deformation theory of a node: the versal deformation space (see [6]) of the completion X_{s} of a nodal curve has dimension dim Ext^{1}(Ω_{X_{s}}^{1}, O_{X_{s}}) = 1, and it is easy to see that the equation uv = t is versal. An elementary proof by lifing the equation is sketched in [Alteration], 2.23.

3.5 Serre’s lemma.

A critical result in the theory of moduli of curves is, that a 1-parameter family of curves admits stable reduction after a base change (see Theorem 11.2). A crucial point in the proof is the relationship between the automorphisms of a curve and the automorphisms of its jacobian, as in the following lemma.

Lemma 3.10. Let C be a stable curve defined over an algebraically closed field k, let m ∈ ℤ_{≥3}, not divisible by the characteristic of k, and let φ ∈ Aut(C) such that the induced map on locally free sheaves of order m

φ_{∗} : Pic_{C}^{0}[m] ⟶ Pic_{C}^{0}[m]

is the identity map. Then φ = 1_{C}, the identity morphism on C.

Proof (see [63], or [16], 3.5.1). Let C̃ → C be the normalization of C (namely the disjoint union of the normalizations of all irreducible components). Denote J := Pic_{C}^{0} and X := Pic0 ; C̃. Consider the “Chevalley decomposition” (as in [14]):

0 → T ⟶ J ⟶ X → 0,

i.e. T ⊂ J is the maximal connected linear subgroup in J, the quotient is an abelian variety, and J/T ≅ Pic0 ; C̃. Note that T ≅ (𝔾_{m})^{s} is a split torus. Define f := φ_{∗} − 1_{J} ∈ End(J). Using

Hom(T, X) = 0

we obtain a commutative diagram

0 → T ⟶ J ⟶ X → 0
g ↓ f ↓ h ↓
0 → T ⟶ J ⟶ X → 0.

By the original lemma of Serre we deduce that h = 0; let us sketch the argument. The automorphism φ is of finite order (because C is stable), hence the induced ψ ∈ Aut(C̃) is of finite order, hence ψ_{∗} = 1 + h is of finite order. Note that the ring End(X) is torsion free, and since ψ is of finite order the subring ℤ[ψ_{∗}] ⊂ End(X) is cyclotomic. By assumption the element ψ_{∗}⊗_{ℤ} ℤ/m = 1 in End(X)⊗_{ℤ} ℤ/m. Since ψ_{∗} is a root of unity, and m ≥ 3, this implies ψ_{∗} = 1_{X}, hence h = 0.

Moreover an analogous reasoning implies that g = 0: use that End(T) = Mat(s, ℤ) is torsion-free. From h = 0 and g = 0 we deduce that f : J → J factors as

J → X ⟶^{f′} T → J.

Using

Hom(X, T) = 0

we conclude f′ = 0, hence f = 0. Hence φ_{∗} = 1_{J}, and this implies that φ = 1_{C}. 3.10

Remark 3.11. We have used the fact that for p ≥ 3, even modulo p, the root of unity ζ_{p} is not equal to 1: indeed, the ring O_{ℚ(ζ_{p})}/p is artinian, with generator ζ_{p}.

3.6 Extending stable curves.

Suppose we are given a stable curve C_{U} → U over an open set U ⊂ S of a base scheme S. Can it be extended to a stable curve C → S? In general the answer is negative. This question is discussed in [32] and [43], where we find criteria which ensure that in certain cases this is possible. The general situation has the following answer: an extension to a stable curves is possible after an alteration on the base. Note the difference from the Flattening Lemma, which has to do with extending families of flat subschemes of a fixed scheme.

Theorem 3.12 (Stable Extension Theorem). Let S be a locally noetherian integral scheme, let U ⊂ S be an dense, open subset and let C → U with sections s_{i}^{U} : U → C be a stable pointed curve. There exists an alteration φ : T → S, and a stable pointed curve D → T with sections τ_{i} : T → D, such that, if we write φ^{−1}(U) =: U′ ⊂ T, we have an isomorphism

D|_{U′} ⟶^{ϕ} U′ ×_{U} C,

such that ϕ^{∗}s_{i}^{U} = τ_{i}.

Remark 3.13. A proof for unpointed curves can be found in [16], Lemma 1.6. We present here a somewhat different proof. For simplicity of notation the proof is stated in the case of unpointed curves.

The first step is to extend isomorphisms of stable curves. The first lemma is the following:

Lemma 3.14. Suppose T is the spectrum of a discrete valuation ring, and D → T and D′ → T are stable (pointed) curves, such that the generic fibres are isomorphic: D_{η} ≅ D′_{η}. Then this extends a unique isomorphism: D ≅_{T} D′.

For the proof see [17], Lemma 1.12. The main point is that the minimal models of D and D′ coincide, and D or D′ are obtained from the minimal model in a unique way by blowing down (−2)-curves. This lemma implies the following (see [17], 1.11):

Lemma 3.15. Suppose T is a scheme, and D → T and D′ → T are stable (pointed) curves. Then Isom_{T} (D, D′) → T is finite and unramified.

Indeed, the previous lemma implies that Isom_{T} (D, D′) → T is proper. Since stable curves have a finite automorphism groups, the morphism is finite. And since stable curves have have no nonzero vector fields, the morphism is unramified.

As a consequence we get the following general result about extending isomorphisms:

Lemma 3.16. Suppose T is an integral normal scheme, and D → T and D′ → T are stable (pointed) curves, such that the generic fibres are isomorphic: D_{η} ≅ D′_{η}. Then this induces an isomorphism: D ≅_{T} D′.

Proof: The given isomorphism over the generic point η gives a lifting η → Isom_{T} (D, D′). The closure of its image in Isom_{T} (D, D′) maps finitely and birationally to T. By Zariski’s Main theorem it is isomorphic to T, and therefore gives a section of Isom_{T} (D, D′) → T.

Exercise 3.17. We show that the condition “normal” in the previous lemma is needed. To this end, choose a regular curve T_{0}, and a smooth curve D_{0} → T_{0}. Choose it in such a way that the geometric generic fibre has only the identity as automorphism, and such that there exist closed points x, y ∈ T_{0} and two different isomorphisms

α, β : (D_{0})_{x} ⟶^{∼} (D_{0})_{y}.

Let T_{0} → T be the nodal curve obtained by identifying x and y as a nodal point P ∈ T (and the curves isomorphic outside these points). Construct D_{α} → T by “identifying (D_{0})_{x} and (D_{0})_{y} via α”. Analogously D_{β} → T. Show that

D_{α} ≅_{T̸} D_{β}, and (D_{α})_{η_{T}} = (D_{0})_{η_{T′}} = (D_{β})_{η_{T}} .

It is instructive to describe Isom_{T} (D_{α}, D_{β}).

Remark 3.18. The phenomenon described in the exercise is characteristic of situations where one has a coarse moduli space rather than a fine one. See Definition 10.7 for details.

The following is an analogous lemma about isomorphisms of the geometric generic fibers:

Lemma 3.19. Suppose T is an integral scheme, D → T and D′ → T stable (pointed) curves, such that the geometric generic fibres are isomorphic:

D\overline{η} ≅ D′ ; \overline{η}.

Then there exists a finite surjective morphism T′ → T and an isomorphism

D ×_{T} T′ ≅_{T′} D′ ×_{T} T′.

Remark 3.20. It is easy to give examples where the isomorphism requested does not exist even over the generic point of T.

Proof: As in [17], 1.10 we consider Isom_{T} (D, D′). The condition in the lemma assures that this is not empty, it is finite and dominant over T, and the lemma follows.

Proof of Theorem 3.12: Here we use the fact that there exists a “tautological family” of curves over the compactified moduli space of curves with a level structure. For stable curves without points, this is given in [21]. For the case of stable pointed curves, use the results in Section 13.4. Another proof, more in the line of [17], is sketched in 13.4

Let us suppose that there exists m ∈ ℤ_{≥3} such that S → Spec(ℤ[1/m]). Hence the family C → U defines a moduli morphism

f : U → \overline{M_{g}}[1/m] := \overline{M_{g}} ×_{Spec ℤ} Spec ℤ[1/m].

We write M := \overline{M_{g}^{(m)}} (after having fixed g and m). We have a curve Z → M such that the associated morphism to \overline{M_{g}}[1/m] is the natural morphism π : M → \overline{M_{g}}[1/m] (we say that Z → M is a tautological family). Let U′′ := U ×\overline{M_{g}} M, let U′ ⊂ U′′ be a reduced, irreducible component of U′′ dominant over U, and let C′ be the pull back C′ = C ×_{U} U′. Let Z′ → U′ be the pull back of the tautological family Z′ = Z ×_{M} U′. By the previous lemma we can replace U′ by a finite cover (call it again U′) for which there is a U′ isomorphism C′ ≃ Z′. Let S′ be the normalization of S in the function field of U′. We define V ⊂ S′ × M to be the image of U′ by the two morphisms into S′ and M, and let T = \overline{V}^{Zar} ⊂ S′ × M.

By construction there is a stable curve over T, obtained by pulling back Z → M, which moreover by construction extends the pull back of C′ → V. This proves the theorem in case S → S_{m} := Spec(ℤ[1/m]).

In case S → Spec(ℤ) is surjective, one does the construction for two different values of m, and then one pastes the result using Lemma 3.19. 3.12

3.7 Contraction and stabilization.

In [39], II, Section 3, pp. 173-179, we find a description of the following two constructions.

1. Consider a stable (n + 1)-pointed curve (X , P) → S with 2g − 2 + n > 0. Deleting one section gives a nodal n-pointed curve (with an extra section), which need not be a stable n-pointed curve. However, if necessary one can contract “non-stable components” of fibres (regular rational curves containing not enough singularities and marked points). After this blowing down one obtains a stable n-pointed curve (X′, Q) → S, and an Smorphism X → X′ mapping the first n sections of P to Q. This process, which arrives at a unique solution to this problem, is called ”contraction”.

2. Consider a stable n-pointed curve (Y, Q) → S plus an extra section σ : S → Y not in Q. This extra section may meet sections in Q, or meet the nodes of Y → S. One can blow up Y in such a way that the strict transforms (in the old sense) of elements of Q and of the extra section give a stable (n + 1)-pointed curve (X , P) → S, and an S-morphism X → Y mapping the first n sections of P to Q. This process, which arrives at a unique solution to this problem, is called ”stabilization”.

4 Proof of de Jong’s main theorem

One striking feature of the proofs of de Jong’s theorem and its derivatives is, that all the ingredients, with the exception of one subtle, but still natural, result (the Three Point Lemma), were known and understood nearly two decades before. The way they are put together is quite ingenious.

4.1 Preparatory steps and observations

The proof of de Jong’s theorem starts with a series of simple reduction steps. The situation. We want to prove de Jong’s Theorem 2.4. Thus we are given a variety X defined over an algebraically closed field k, and a Zariski - closed subset Z ⊂ X. We now perform some elementary reductions: Replacing X by an alteration. In order to prove the theorem for a varety X and a closed subset Z, it is enough to prove it for an alteration X′ of X while replacing Z by its inverse image Z′ in X′. Thus in several stages of the proof, once we find an alteration X′ → X which we like better than X, we simply replace the pair (X, Z) by (X′, Z′). Making Z into a divisor. By blowing up Z in X, and using the observation above, we may assume that Z is the support of an effective Cartier divisor. We will slightly abuse terminology, and say that “Z is a divisor” when we mean that Z is a closed subset supporting an effective Cartier divisor. Enlarging Z. Suppose Z_{i} ⊂ X are divisors and Z_{1} ⊂ Z_{2}, then to prove the theorem for (X, Z_{1}) it suffices to prove it for (X, Z_{2}). Indeed, if f : Y → X is an alteration such that Y is nonsingular and f^{−1}(Z_{2}) is a strict normal crossings divisor, then f^{−1}(Z_{1}) is a Cartier divisor contained in f^{−1}(Z_{2}), and it is clear from the definition that it i a strict normal crossings divisor as well. Thus we may always enlarge the divisor Z. Making X quasi-projective. Using Chow’s Lemma 3.3, we may assume X is quasi-projective. Indeed, by Chow’s lemma there is a modification X′ → X such that X′ is quasi-projective. We may replace X by X′. Enlarging X. Suppose X ⊂ X_{1} is an open embedding of varieties, Z_{1} ⊂ X_{1} a divisor which containins X_{1} X, and Z = X ∩ Z_{1}. Then evidently to prove the theorem for (X, Z) it suffices to prove it for (X_{1}, Z_{1}). Making X projective. Since X is quasi-projective, there is an open embedding X ⊂ \overline{X} where \overline{X} is projective. Denote Z_{1} = \overline{Z} ∪ (\overline{X} X). We may replace \overline{X} by the blowup of Z_{1}, thus we may assume that Z_{1} is the support of a Cartier divisor. By the previous observaton it is enough to prove the result for (\overline{X}, Z_{1}). We may assume X is normal. Indeed, we can simply replace X by its normalization.

To summarize, one may assume that the variety X is projective and normal, and the subset Z is the support of an effective Cartier divisor. Moreover, one may always replace Z by a larger divisor.

4.2 Producing a projection

The next step is to produce a projection with some nice properties. We first start with some general facts about projections in projective spaces.

Let Y ⊂ ℙ^{N} be a projective variety over an algebraically closed field (in fact, separably closed would suffice). For any closed point p ∈ ℙ^{N} Y we have a projection pr_{p} : Y → ℙ^{N−1}.

Lemma 4.1. Suppose dim Y < N − 1. Then there is a nonempty open set U ⊂ ℙ^{N}, such that if p ∈ U then pr_{p} sends Y birationally to its image.

Proof. Let q ∈ Y be a regular point. Define the cone C_{Y,q} over Y with vertex q to be the Zariski closure of the union of all secant lines lines containing q and another q′, for all q′ ∈ Y. It is easy to see that C_{Y,q} has dimension ≤ dim Y + 1 < N. Note that C_{Y,q} containes (as “limit points”) the projective tangent space 𝕋_{Y,q} at q. Therefore if p ∈ ℙ^{N} C_{Y,q} then the line through p and q meets Y transversally, at q only. This property holds as well for the line through p and q′, for any q′ ∈ Y in a neighborhood of q. Hence the lemma.

Lemma 4.2. Suppose dim Y = N − 1. Then there is a nonempty open set U ⊂ ℙ^{N}, such that if p ∈ U then pr_{p} maps Y generically étale to ℙ^{N−1}.

Proof. Same as before, using 𝕋_{Y,q} instead of C_{Y,q}.

We now go back to our X and Z.

Lemma 4.3. There exists a modification ϕ : X′ → X and a morphism f : X → ℙ^{d−1} such that

1. There exists a finite set of nonsingular closed points S ⊂ X_{ns} disjoint from Z, such that X′ is the blowup of X at the points of S.

2. f is equidimensional of relative dimension 1

3. The smooth locus of f is dense in all fibers

4. Let Z′ = ϕ^{−1}Z. Then f|_{Z′} is finite and generically étale

5. At least one fiber of f is smooth.

Proof. First project π : X → ℙ^{d} using the previous lemmas N − d − 1 times.

Let B ⊂ ℙ^{d} be the locus over which π is not étale.

If we choose a general p ∈ ℙ^{d}, then pr_{p} : π(Z) → ℙ^{d−1} is generically étale - simply use the lemma above for all irreducible components of π(Z).

We choose such a p away from B. By the local description of blowing up, we can identify the variety

X′ = {(x, ℓ) ∈ X × ℙ^{d−1}|π(x) ∈ ℓ}

with the blowing up of X at the points in π^{−1}(p).

We define f : X′ → ℙ^{d−1} to be the second projection.

We can identify the fibers: the fiber over a point ℓ is the scheme theoretic inverse image π^{−1}(L) where L is the line corresponding to ℓ.

It follows immediately that f is equidimensional: all fibers have dimension at most 1, and are defined by d − 1 equations (the equations of L).

Since no line through p is contained in B, every fiber has a dense smooth locus.

The last assertion follows by Bertini’s theorem, since the fibers are obtained by intersecting X with linear subspaces.

Lemma 4.4. The morphism f has connected fibers.

Proof. Since the smooth locus is dense in every fiber, the Stein factorization is étale. Since projective space has no nontrivial finite étale covers, the Stein factorization is trivial.

Remark 4.5. 1. The last assertion is not really necessary: if f did not have connected fibers, we could replace f : X′ → ℙ^{d−1} by its Stein factorization.

2. The projection above is the only point where it is crucial that X should be normal, to guarantee that the generic fiber is smooth. From here on we will allow ourselves to make reductions after which X might not be normal.

To summarize, one may assume that we have a morphism of varieties X → P, for some variety P, which makes X into a generically smooth family of curves, satisfying some nice properties, in particular Z → P is finite and generically étale.

4.3 Enlarging the divisor Z

In order to “rigidify” the situation, it will be useful to enlarge Z so it meets every fiber “sufficiently”. This is done as follows:

Lemma 4.6. Let X → P be as above. There exists a divisor H ⊂ X such that

1. f|_{H} : H → P is finite and generically étale, and

2. for any irreducible component C of a geometric fiber of f, we have

#sm(X/P) ∩ C ∩ H ≥ 3.

Here we count the points without multiplicities.

Proof. Let n ≥ 1 be an integer. Given a very ample line bundle L on X, consider the embedding

i : X ↪ ℙ = ℙ(Γ(X, L^{⊗n}))

associated to L^{⊗n}.

Claim. Given any irreducible curve C ⊂ X, the image i(C) ⊂ ℙ is not contained in any linear subspace of dimension n − 1.

Proof of claim. Since L is very ample, the image of Γ(X, L) → Γ(C, L_{|C}) containes a base point free pencil V ⊂ Γ(C, L_{|C}). The map Sym^{m}V → Γ(C, L_{|C}^{⊗n}) has rank ≥ n + 1, therefore Γ(X, L^{⊗n}) → Γ(C, L_{|C}^{⊗n}) has rank ≥ n + 1, which is what we claimed. (Claim)

We consider the collection of “bad” divisors and show that there are “good” ones left. So consider

T = {(H, y) ∈ ℙ^{∨} × P | dim f^{−1}y ∩ H = 1} ⊂ ℙ^{∨} × P.

It is clear that T is a Zariski closed subset. We can describe the fibers of pr_{2} : T → P using irreducible components of the fibers:

pr_{2}^{−1}(y) = ⋃_{C⊂f^{−1}y}{H|i(C) ⊂ H}.

But by the fact that i(C) is not contained in any linear subspace of dimension n − 1, we have

codim(pr_{2}^{−1}(y), ℙ^{∨}) ≥ n.

Therefore dim T ≤ dim P + dim ℙ^{∨} − n.

Thus if n is large enough, pr_{1}(T) ⊂ ℙ^{∨} is of large codimension (at least n − dim P). In particular pr_{1}(T) ≠ ℙ^{∨}.

We now fix such large n. Thus there are plenty of H which map finitely to P.

For a fixed closed point y ∈ P (k) consider the set

U(y) = \{H ∈ ℙ^{∨}(k) | \begin{matrix}H∉pr_{1}(T) \\ H ∩ f^{−1}y ⊂ sm(X/P) \\ H ∩ f^{−1}y is reduced \end{matrix} \} .

This is clearly a nonempty open set of ℙ^{∨}. Moreover, if H ∈ U(y) then H ∈ U(y′) for all y′ in a neighborhood of y.

If moreover n ≥ 3, then we have that #H ∩ f^{−1}y ≥ 3. so we are done for all points in a neighborhood V of y.

We deal with points in P V in the same way. Using Noetherian induction we are done. (Lemma)

4.4 The idea of simplifying the fibers

De Jong’s idea is to simplify the fibers of the morphism X → P. Then by induction on dimension one can simplify the base P, and finally put these simplifications together.

The method of simplifying the fibers uses the deepest ingredient in the program: the theory of moduli of curves (see Section 10 for discussion).

Here is the general plan. First, as we will see below, it is easy to make an alteration of P, and replace X and Z by their pullbacks, such that Z becomes the union of sections of X → P.

We can think of the generic fiber of X → P as a smooth curve with a number of points marked on it. Say the genus of this curve is g, and the number of points is n. By the Stable Extension Theorem 3.12, the generic fiber can be extended, after an alteration P_{1} → P, to a family of stable curves X_{1} → P_{1}:

\begin{matrix}X_{1} & −−→ & X \\ ↓ & & ↓ \\ P_{1} & → & P \end{matrix}

The new morphism f_{1} : X_{1} → P_{1} is much nicer than f, since at least the fibers are as nice as one can expect: they are nodal curves. Moreover, Z was made much nicer: it is replaced by n sections which are mutually disjoint, and pass through the smooth locus of f_{1}.

If we can resolve P_{1} (say using induction on dimension), then it is easy to resolve X_{1} as well.

There is a problem though: if we want to repeat this inductively, we cannot allow a rational map X_{1} −−→ X which is not a morphism - since we cannot pull back nicely along rational maps. So we want to find a way to make sure that X_{1} −−→ X is actually a regular map.

Remark 4.7. If one is satisfied with proving a weaker result, namely that every variety admits a “rational alteration” by a nonsingular variety, then there is an alternative way to avoid the issue. This is carried out by S. Mochizuki in [43].

Remark 4.8. Another way to circumvent the issue of extending β to a morphism, is to ensure that it extends automatically, by using a moduli space into which a morphism is built in: the space of stable maps. This was carried out, in characteristic 0, in [1], Lemma 4.2. Unfortunately the details of constructing moduli spaces of stable maps have not yet been written out in positive or mixed characteristics, although this would not be difficult: the results of [9] imply that the moduli of stable maps forms a proper Artin stack, and the results of [37] imply that this stack admits a proper algebraic space as a coarse moduli space. One should even be able to modify the argument of [40], Proposition 4.5 and show that this space is projective, but this is not essential for the argument.

Let us go into details.

4.5 Straightenning out Z.

Lemma 4.9. There exists a normal variety P_{1} and a separable finite morphism P_{1} → P satisfying the following property:

Let X_{1} = X̃_{P_{1}} be the strict transform (see Definition 3.1), and let Z_{1} be the inverse image of Z in X_{1}. Then there is an integer n ≥ 3, and n distinct sections s_{i} : P_{1} → X_{1}, i = 1, …, n such that

Z_{1} = ⋃_{i=1}^{n} s_{i}(P_{1}).

Proof. This can be proven by induction on the degree n of Z → P as follows: Let Z_{1} be an irreducible component of Z and let P′ := Z_{1}^{ν} be its normalization. We have a generically étale morphism P′ → P. Denote X′ = X̃_{P′} and let Z′ be the inverse image of Z. The morphism P′ → Z gives rise to a section s_{k+1} : P′ → Z′, and therefore we can write Z′ = s_{k+1}(P′) ∪ Z′′. We have deg(Z′′ → P′) = deg(Z → P) − 1, and therefore the inductive assumption holds for Z′′.

4.6 Producing a family of stable pointed curves.

Let X → P, s_{i} : P → X be the new family. Let U ⊂ P be an open set satisfying the following assumptions:

1. X_{U} → U is smooth;

2. the sections s_{i}|_{U} : U → X_{U} are disjoint.

Such an open set clearly exists.

Since n ≥ 3 this gives the morphism X_{U} → U the structure of a family of stable n-pointed curves.

And here comes the point where moduli theory is used: by Theorem 3.12, there exists an alteration P_{1} → P, a family of stable pointed curves C → P_{1}, with sections τ_{i} : P_{1} → C, such that over the open set U_{1} = P_{1} ×_{P} U ⊂ P_{1} we have an isomorphism sm β : C_{U_{1}} → U_{1} ×_{P} X, satisfying β^{∗}s_{i} = τ_{i}.

4.7 The three point lemma.

As usual, we replace P by P_{1} and X by its strict transform. Thus we may assume that we have a diagram as follows:

\begin{aligned}C & −−→^{β} & X \\ & ↘ & ↓ \\ & & P \end{aligned}

The crucial point, for which we needed to “enlarge Z” in a previous step, is the following:

Lemma 4.10 (Three Point Lemma). Suppose Z meets the smooth locus of every irreducible component of every fiber in at least three points. Then, at least after a modification of P, the rational map β : C −−→ X extends to a morphism.

The proof of this lemma, which is detailed in the next few paragraphs, is probably the most subtle point in this chapter.

4.8 Flattening the graph.

Let T ⊂ X ×_{P} C be the closure of the graph of the rational map β. We have two projection maps pr_{1} : T → C and pr_{2} : T → X.

Claim. There exists a modification P′ of P such that the strict transform of X, and the closure of the graph of C −−→ X are both flat. Thus we might as well assume X → P and T → P are flat.

Proof. By the Flattening Lemma 3.4 there exists a modification P′ → P such that X̃_{P′} and T̃_{P′} are both flat. Evidently the closure of the graph of the rational map C ×_{P} P′ → X̃_{P′} is contained in T̃_{P′}, and since T̃_{P′} is flat they coincide by [HAG] III.9.8. (Claim)

Let p be a geometric point on P, and denote by X_{p}, T_{p}, C_{p} the fibers over p. There exists a finite set W ⊂ X_{p} such that T_{p} → X_{p} is finite away from W. Indeed, the flatness implies that dim T_{p} = dim X_{p} = 1.

Thus, for any x ∈ X_{p} W, there is an open neighborhood x ∈ V ⊂ X such that pr_{2}^{−1}V → V is finite and birational.

In case x ∈ Sm(X_{p}) W, we may choose V ⊂ Sm(X → P). Using the assumption that P is normal, it follows that V is normal. In this case, by Zariski’s main theorem, pr_{2}^{−1}V → V is an isomorphism.

Note that the assumption that x ∈ Sm(X_{p}) excludes only finitely many points, since our projection X → P is smooth at the generic point of each component of the geometric fiber X_{p}. Therefore we conclude that the following lemma holds:

Lemma 4.11. If X′ ⊂ X_{p} is an irreducible component, then there is a unique irreducible component T′ of T_{p} mapping finitely onto X′ via pr_{2} : T → X. Moreover, T′ → X′ is birational.

Repeating the argument for pr_{1} : T → C, we also have:

Lemma 4.12. If C′′ ⊂ C_{p} is an irreducible component, then there is a unique irreducible component T′′ of T_{p} mapping finitely onto C′′ via pr_{1} : T → C. Moreover, T′′ → C′′ is birational.

Let x ∈ Sm(X_{p}) be a closed point. Considering the Stein factorization T → X̃ → X, we have that the fiber pr_{2}^{−1}(x) is connected. Indeed, since X is normal at x, we have that X̃ → X is an isomorphism at x.

4.9 Using the three point assumption

Let X′ ⊂ X_{p} be an irreducible component, and T′ ⊂ T_{p} the unique component mapping finitely (and birationally) onto it, as in Lemma 4.11 above. We will prove that pr_{1} : T′ → C is non-constant. Assume by contradiction that pr_{1}(T′) = c is a point.

We will now use the three point assumption. Let s_{i} : P → X, i = 1, …, 3 be three of the given sections such that s_{i}(p) = x_{i} are three distinct points on Sm(X′). Let T_{i} = pr_{2}^{−1}x_{i}. Let τ_{i}(p) = c_{i} ∈ C_{p}.

Note that the point t_{i} = (c_{i}, x_{i}) ∈ C × X is in T.

Assume c ∉ {c_{i}, i = 1, …, 3}. Then each of T_{i}, i = 1, …, 3 contains an irreducible component T′_{i} whose image in C is again a curve passing through c. These image components are distinct. Indeed, T_{i} are disjoint subschemes of T_{p}, whose images in C conect c with c_{i}, and therefore each has an irreducible component whose image contains c. These components are distinct, and by Lemma 4.12 their images are distinct.

This contradicts the fact that C_{p} is nodal. Thus c is among the c_{i}.

Assume now, without loss of generality, c = c_{1}. Repeating the argument of the previous paragraph we conclude that there are two distinct components of C_{p} passing through c. This contradicts the fact that C_{p} has a marked point at c = c_{1}.

Thus we conclude that pr_{1} : T → C is finite and birational.

By Serre’s criterion C is normal: it is clearly regular in codimension 1, and condition S_{2} follows since C → P has reduced one-dimensional fibers and P is normal.

We conclude that T → C is an isomorphism, hence β extends as a morphism! 4.10

4.10 Induction

We arrived at the following situation:

\begin{aligned}C & →^{β} & X \\ & ↘ & ↓ \\ & & P \end{aligned}

We may replace X by C, and Z by its inverse image in C. Note that Z is no longer finite over P: it has a“finite part”, the union of the sections τ_{i} : P → C = X, but there is a “vertical” part Z_{vert}, which is the union of irreducible components of singular fibers of X → P.

Let Σ ⊂ P be the closed subset over which f : X → P is not smooth. By the induction assumption there is a projective alteration P_{1} → P such that P_{1} is nonsingular and the inverse image of Σ is a strict normal crossings divisor. We may replace X by its pullback to P_{1}, and replace P by P_{1}. It is convenient to replace Z by its union with f^{−1}(Σ).

We arrived at a situation where both P, and the morphism f : X → P, are simplified. The resulting variety has very simple singularities, and its desingularization results from the following exercises.

4.11 Exercises on simple blowup of nodal families.

Let f : X → S be a flat morphism of varieties over an algebraically closed field k, with n = dim X = dim S + 1. Let D ⊂ S be a reduced divisor. We make the following assumptions.

N1 The base S is nonsingular.

N2 The divisor D has strict normal crossings.

N3 The morphism f is smooth over S D.

N4 The morphism f : X → S is a nodal curve.

Let x ∈ X be a closed point and let s = f(x) ∈ S. By assumption we may choose a regular system of parameters t_{1}, ..., t_{n−1} at s such that D coincides on a neighborhood with the zero locus of t_{1} ⋯ t_{r} for some r ≤ n − 1. It can be seen that if x is a singular point of X, then the completed local ring of X at x can be described as

(∗) k[[u, v]]/(uv − t_{1}^{n_{1}} · · · · · t_{r}^{n_{r}})

.

Step 1: Assume codim_{X} Sing(X) = 2.

1. Show that there is an irreducible component D_{1} ⊂ D and Σ_{1} ⊂ Sing(X) such that f(Σ_{1}) = D_{1}.

2. Fix a point x ∈ Σ_{1}, and use formal coordinates as in (∗), such that D_{1} = V (t_{1}). Show that the power n_{1} of t_{1} is > 1.

3. Show that the ideal of Σ_{1} in the formal completion is (u, v, t_{1}).

4. Conclude that Σ_{1} → D_{1} is étale, in particular Σ_{1} is nonsingular.

5. Let X_{1} = Bl_{Σ_{1}}X. Show that X_{1} → S satisfies conditions N1-N4, there is at most one component of Sing(X_{1}) over Σ_{1}, with the exponent n_{1} replaced by n_{1} − 2

6. Conclude by induction that there is a blowup X′ → X centered above Sing(X), such that X′ satisfies N1-N4, and codim_{X} Sing(X) > 2.

7. Show that each component of Sing(X′) is defined by u = v = t_{i} = t_{j} in equation (∗), in particular it is nonsingular.

Step 2: Assume codim_{X} Sing(X) > 2. Define Z = f^{−1}D. Unfortunately here we need to abandon the structure X → B of a family of nodal curves. Instead we look at X itself. The situation is as follows:

T1 whenever x is a nonsingular point of X, Z has normal crossings at x.

T2 whenever x ∈ Sing(X), we have formal description

(∗∗) k[[u, v]]/(uv − t_{1} ⋯ t_{s}), 2 ≤ s ≤ r ≤ n − 1
and Z = V (t_{1} ⋯ t_{r}).

T3 All components of Sing(X) are nonsingular.

1. Let E ⊂ Sing(X) be an irreducible component. Show that the blowup Bl_{E}X satisfies T1-T3, and its singular locus has one fewer irreducible component.

2. Conclude by induction that there is a resolution of singularities X′ → X.

This concludes the proof of Theorem 2.4

5 Modifications of the proof for Theorems 2.5 and 2.8.

5.1 Exercises on removing the conditions on the projection.

An important step in the proof of de Jong’s theorem was, that given the projection X → P, one could construct an alteration P_{1} → P and a diagram

\begin{matrix}C & →^{β} & X \\ ↓ & & ↓ \\ P_{1} & → & P \end{matrix}

where C → P_{1} was a family of nodal curves. In order for the proof to go through, we made several assumptions on the projection X → P. Here we will show that even if these conditions fail, we can still reduce to the case where they do hold.

Exercise 5.1. Using an alteration, show that the condition that Z → P be finite in Lemma 4.3 (4) is unnecessary for the rest of the proof.

Exercise 5.2. Show that, if one is willing to accept inseparable alterations in the theorem, the condition that Z → P be generically étale in Lemma 4.3 (4) is unnecessary for the rest of the proof.

Exercise 5.3. ∗ By reviewing the arguments, show that the condition that every component of every fiber of X → P be generically smooth is unnecessary.

Here a modification of the three point lemma is be necessary! In [31], de Jong uses a trick of “raising the genus of the curves” with finite covers. Another way goes as follows: in the proof of the Three Point Lemma 4.10, after flatenning X and T, one works with fibers of the normalizations X^{ν} and T^{ν}. This way one avoids the need for Sm(X_{p}) to be dense. One notes that the sections s_{i} lift to X^{ν}, and at least three meet every component of every fiber, since Z is the support of a Cartier divisor! The details are left to the reader.

Exercise 5.4. Using the flattening lemma and the previous exercise, show that the condition that X → P be equidimensional is unnecessary.

Exercise 5.5. Show that, if one is willing to accept inseparable alterations in the theorem, the condition that the generic fiber of X → P be smooth is unnecessary.

5.2 Exercises on Theorem 2.5.

Let us address Theorem 2.5 on semistable reduction up to alteration. Suppose S = Spec R where R is a discrete valuation ring, X → S is a morphism as in the theorem, and Z a proper closed subset.

Exercise 5.6. Show that we may assume X projective over S, that Z is the support of a Cartier divisor.

Exercise 5.7. Show that we may assume that the generic fiber is a normal variety, and that X is a normal scheme. (You may need an inseparable base change!)

Exercise 5.8. Let d be the dimension of X_{η}. Produce a projection X → ℙ_{S}^{d−1} with connected fibers.

Exercise 5.9. Use the semistable reduction argument, with the Three Point Lemma, and the results of Section 5.1 to replace X by a nice family of curves X → P → S.

Exercise 5.10. Use induction on the dimension to conclude the proof of the theorem.

Exercise 5.11. Can you think of other situations where a similar theorem can be proven, where S is not necessarily the spectrum of a discrete valuation ring? (This is interesting even in characteristic 0!)

5.3 Exercises on theorem 2.8

We now address the equivariant version of the theorem. Suppose X is a variety, Z a proper closed subset and G acts on X stabilizing Z. We wish to prove Theorem 2.8.

Exercise 5.12. Produce an equivariant version of Chow’s lemma, so that we may assume X is projective.

Exercise 5.13. Show that, to prove the theorem, it suffices to consider teh case where Z is a divisor.

Exercise 5.14. Show that we may replace Z by a bigger equivariant divisor; in particular we may assume Z contains the fixed point loci of elements in G.

Exercise 5.15. Using a projection of X/G, show that we may assume we have an equivariant projection X → P making X into a nice family of curves.

Exercise 5.16. Consider the case X = 𝔸_{k}^{2} where char k = p, and G = ℤ/p acting via (x, y) ↦ (x, x + y). Show that the fixed point set maps inseparably to the image. In particular, the map Z → P in the previous exercise might be inseparable!

Exercise 5.17. Making an alteration “Galois”: Given a variety W, a finite group action H ⊂ Aut W and an alteration V_{0} → W, show that there exists an alteration V → V_{0}, and a finite group H′ with a surjection H′ → H, and a lifting of the H action H′ ⊂ Aut(V → W) such that the extension of fixed fields K(W)^{H} ⊂ K(V)^{H′} is purely inseparable.

Exercise 5.18. ∗ Use the uniqueness in the stable reduction theorem to show that there is an alteration P′ → P, a family of stable pointed curves C → P′ and a finite group G′ with a quotient G′ → G and a diagram

\begin{aligned}C & → & X \\ ↓ & & ↓ \\ P′ & → & P \end{aligned}

on which G′ acts equivariantly, such that C is birational to X̃_{P′}, and the extensions K(X)^{G} ⊂ K(C)^{G′} and K(P)^{G} ⊂ K(P′)^{G′} are purely inseparable.

Exercise 5.19. Use induction on the dimension and a suitable modification of the elementary blowups argument to conclude the theorem.

6 Toroidal geometry

6.1 Basic definitions.

For simplicity we work over an algebraically closed field. We recall the notion of a toric variety (a more thorough discussion is available in [15]):

Definition 6.1. A variety X together with an open dense embedding T ⊂ X is called a toric variety if X is normal, T is a torus (geometrically isomorphic to 𝔾_{m}^{k}), and the action of T on itself by translations extends to an action on X.

To get an intuitive idea about the singularities of a toric variety, it is worth noting that a normal, affine variety, defined by equations between monomials is toric, and every toric variety is locally of this type.

For many purposes toric varieties are too restrictive. A more general notion was introduced by Mumford in [38]:

Definition 6.2. A variety X together with an open embedding U ⊂ X is called a toroidal embedding if any point x ∈ X has an étale neighborhood X′ such that X′ ×_{X} U ⊂ U is isomorphic to an étale neighborhood of a point on a toric variety.

Thus a toroidal embedding looks locally like a toric variety, and the big open set U is a device which ties together these “local pictures”. In a sense, this notion is suitable for studying varieties whose singularities are like those of toric varieties.

In this section we recall facts about toric varieties and briefly indicate how one can obtain analogous facts about toroidal embeddings. The details are available in [38].

Remark 6.3. A more sheaf - theoretic approach was introduced by K. kato, see [35], [36].

Definition 6.4. A toroidal embedding is said to be strict (or a toroidal embedding without self intersections) if every irreducible component of X U is normal.

For instance, if X is a nonsingular variety, D ⊂ X is a strict normal crossings divisor, and U = X D, then U ⊂ X is a strict toroidal embedding.

We will only work with strict toroidal embeddings.

Definition 6.5. If G ⊂ Aut(U ⊂ X) is a finite group, we say that G acts toroidally if for any point x ∈ X, the stabilizer G_{x} of x can be identified with a subgroup of the torus in an appropriate étale neighborhood of x.

Definition 6.6. A morphism between toric varieties is called a toric morphism, if it is surjective and torus-equivariant. A morphism of toroidal embeddings (U_{X} ⊂ X) → (U_{Y} ⊂ Y) is called a toroidal morphism if locally on X it looks like a toric morphism.

6.2 The cone.

First recall some notation (see [15]):

M = Hom(T, 𝔾_{m}) - this is the group of algebraic characters of T; M_{ℝ} = M ⊗ ℝ N = Hom(M, ℤ) = Hom(𝔾_{m}, T) - this is the group of 1-parameter sub-

groups on T; N_{ℝ} = N ⊗ ℝ It is common to call the functions defined by elements of M the monomials.

One uses the notation x^{m} for the monomial associated with the element m ∈ M. Recall the basic correspondence between

{affine toric varieties T ⊂ X} and {strictly convex rational polyhedral cones σ ⊂ N_{ℝ}}

which can be defined in one direction via

V_{σ} = Spec k[σ^{∨} ∩ M],

and in the other direction by

σ = the cone spanned by the 1-parameter subgroups ϕ : 𝔾_{m} → T such that the limit lim_{z→0} ϕ(z) exists in V.

There is another, less well known characterization of σ, which is less dependent on the torus action, and is therefore useful for toroidal embeddings:

Any monomial m ∈ M defines a Cartier divisor Div(x^{m}) supported on V T. If σ^{∨} contains a line through the origin, then for any m on this line the divisor is easily seen to be trivial (both m and −m give regular functions).

We use the following notation: σ^{⊥} = {m ∈ M_{ℝ}|⟨m, σ⟩ = 0} M^{σ} = Cartier divisors supported on V T. One can easily see that M^{σ} = M/σ^{⊥} ∩ M. N_{σ} = span(σ). Clearly N_{σ} = Hom(M^{σ}, ℤ). Let M_{+}^{σ} ⊂ M^{σ}: the effective Cartier divisors. We have that M_{+}^{σ} = σ^{∨} ∩ M/σ^{⊥} ∩ M. It is not hard to see that σ = (M_{+}^{σ})_{ℝ}^{∨}, the dual cone of the cone spanned

by M_{+}^{σ}.

In short: σ is the dual cone to the cone of effective Cartier divisors supported on V T.

6.3 The toroidal picture

We wish to mimic the construction of the cone in the toroidal case. We follow [38], Chapter II.

Let U ⊂ X be a strict toroidal embedding. X U = ∪D_{i}, where D_{i} normal. We decompose ∩_{i∈I}D_{i} = ∪X_{α}; the locally closed subsets X_{α} are called

strata.

Each stratum has its star: Star(X_{α}) = ∪_{X_{α}⊂\overline{X_{β}}}X_{β}.

Note: X_{α} is the unique closed stratum in Star(X_{α}). In a sense it is analogous to the unique closed orbit in an affine toric variety.

Define:

M^{α} = group of Cartier divisors supported on Star(X_{α}) U;

M_{+}^{α} = subset of effective Cartier divisors;

N_{α} = Hom(M_{α}, ℤ);

σ_{α} = (M_{+}^{α})_{ℝ}^{∨}.

Thus, to each stratum we associated a strictly convex rational polyhedral cone.

Remark 6.7. The cone σ_{α} has a description analogous to the toric one using 1-parameter subgroups, in terms of valuations. Let RS(X) be the discrete valuations on X. Let v be a valuation centered in Star(X_{α}). Let f_{j} be rational function defining generators of M_{α} on a small affine open. Then v(f_{j}) is a vector in σ_{α}, and in fact σ_{α} can be described as a set of equivalence classes of discrete valuations centered in Star(X_{α}), the equivalence being defined by equality of the valuations of these fuinctions f_{j}.

6.4 Birational affine morphisms.

Recall: if τ ⊂ σ are two strictly convex rational polyhedral cones, then τ^{∨} ⊃ σ^{∨} gives rise to a morphism V_{τ} → V_{σ}, which is birational and affine.

Note that V_{τ} → V_{σ} can be described in the following invariant manner:

V_{τ} = Spec_{V_{σ}} ∑_{E∈M_{+}^{τ}} O_{V_{σ}}(−E),

where the sum is taken inside the field of rational functions of V_{σ}.

This clearly works over Star(X_{α}) in the toroidal case as well.

6.5 Principal affine opens.

If m ∈ σ^{∨} then τ = {n ∈ σ| < n, m >= 0} is a face of σ. We have τ^{∨} = σ^{∨}+ℝ·m, and therefore V_{τ} is the principal open set on V_{σ} obtained by inverting the monomial x^{m}.

Again, this can be discribed divisorially in terms of Div(m). Thus the same is true for Star(X_{α}): given a face τ of σ_{α}, we get an open set

Star(X_{β}) ⊂ Star(X_{α})

such that τ = σ_{β}.

The most important face of a cone is the vertex. It corresponts to the open set T ⊂ V. In the toroidal case you get U.

6.6 Fans and polyhedral complexes.

Recall: if σ_{1} and σ_{2} intersect along a common face τ, then V_{σ_{1}} and V_{σ_{2}} can be glued together along the common open set V_{τ}, forming a new toric variety.

In general, whenever you have a fan Σ in N, namely a collection of cones σ_{i} intersecting along faces, you can glue together the V_{σ_{i}} and get a toric variety V_{Σ}.

It is not hard to see that every toric variety is obtained in this way in a unique manner. The point is that every toric variety is covered by affine open toric varieties.

In the toroidal case, X is covered by the open sets {Star(X_{α})}_{α}.

In general

Star(X_{α}) ∩ Star(X_{β}) = ∪ Star(X_{γ_{i}}),

so σ_{γ_{i}} are possibly several faces of both σ_{α} and σ_{β}.

Still these can be glued together, as a rational conical polyhedral complex. The main difference from the toric case, is that it is abstractly defined, and in general it is not linearly contained in some vector space N_{ℝ}.

6.7 Modifications and subdivisions.

Let Σ be a fan, and Σ′ → Σ a (complete) subdivision. This corresponds to a toric modification V_{Σ′} → V_{Σ}.

Since the construction is local (the Spec construction, as in Section 6.4, and gluing) it works word for word in the toroidal case. There is a small issue in checking that the resultin modification is still a strict toroidal embedding; this is discussed in detail in [38].

In [38] (see also [15]) it is shown that a modification is projective if and only if the subdivision is induced by a support function - one associates to a support function an ideal, whose blowup gives the modification. This works in the toroidal case as well.

6.8 Nonsingularity

Recall: an affine toric variety V_{σ} is nonsingular if and only if σ is simplicial, generated by a basis of N_{σ} (namely, part of a basis of N). Such a cone is called nonsingular.

In general: a toric variety V_{Σ} is nonsingular if and only if every cone σ ∈ Σ is nonsingular.

This is a local fact, so it is true in the toroidal case as well.

6.9 Desingularization

Recall that it is easy to resolve singularities of a toric variety: one finds a simplicial subdivision such that every cone is nonsingular.

Obviously, the same works in toroidal case! We obtained:

Theorem 6.8. For any toroidal embedding U ⊂ X there is a projective toroidal modification U ⊂ X′ → X such that X′ is nonsingular.

See [38], Theorem 11*, page 94.

6.10 Exercises on toric varieties and toroidal embeddings.

1. Show that 𝔾_{m}^{n} ⊂ 𝔸^{n} is a toric variety. Describe its cone.

2. Show that 𝔾_{m}^{n} ⊂ ℙ^{n} is a toric variety. Describe its fan.

3. Let X ⊂ 𝔸^{n} be a normal variety defined by monic monomial equations of

type

∏ x_{j}^{n_{j}} = ∏ x_{j}^{m_{j}}.

Show that X is toric. (Identify the torus!)

4. Do the same if the monomial equations are not necessarily with coefficients = 1.

5. Describe the cone associated to the affine toric variety defined by

xy = t_{1}^{k_{1}} ⋯ t_{r}^{k_{r}}.

6. Look at the affine 3-fold xy = zw. Let X′ → X be the blowup of X at the ideal (x, z). Describe this blowup, show that it is toric, and describe the cone subdivision associated to it.

7. Let X = 𝔸^{2}, D = {xy(x + y − 1) = 0}, U = X D. Show that U ⊂ X is a toroidal embedding. Describe its conical polyhedral complex. (Compare with the fan of ℙ^{2}!)

8. Do the same for D = {y(x^{2}+y^{2}−1) = 0}. Show that the resulting complex can not be linearly embedded in a vector space.

9. Consider the surface X = {z^{2} = xy}, U = {z ≠ 0}. Show that U ⊂ X is toric and describe its cone.

10. Consider the surface X above, let D_{1} = {x = 0}, D_{2} = {y = x(x − 1)^{2} and z = x(x − 1)}. Let U = X (D_{1} ∪ D_{2}). Show that U ⊂ X is toroidal. Describe its conical polyhedral complex. Make sure to describe the integral structure!

6.11 Abhyankar’s lemma in toroidal terms.

Abhyankar’s lemma about fundamental groups (see [25], [26]) describes the local tame fundamental group of a variety around a normal crossings divisor. Let X = Spec k[[t_{1}, …, t_{n}]] and let D = V (t_{1} ⋯ t_{n}). Let Y → X be a finite alteration which is tamely branched along D, and étale away from D. For m prime to char k, denote X_{m} = Spec k[[t_{1}^{1/m}, …, t_{n}^{1/m}]]. Abhyankar’s lemma says that the normalization of Y ×_{X} X_{m} is étale over X_{m}.

In the following exercises we interpret this in toroidal terms.

Exercise 6.9. Let U ⊂ X be a nonsingular strict toroidal embedding. Let f : Y → X be a finite cover, which is tame, and étale over U. Then f^{−1}U ⊂ Y is a strict toroidal embedding.

Exercise 6.10. Suppose further that Y → X is Galois, with Galois group G. Show that G acts tortoidally on Y.

7 Weak resolution of singularities I

Given the existence of toroidal resolution, the proof of weak resolution of singularities in characteristic 0 by Bogomolov and Pantev is arguably the simplest available. It does not even require surface resolution, or moduli spaces.

We will now go through this proof. The steps of proof here include some simplifications on the arguments in [12], which came up in discussions with T. Pantev. These and additional simplifications were discovered independently by K. Paranjape [52], and we have used his exposition in some of the following exercises.

7.1 Projection.

Let X be a variety over an algebraically closed field of characteristic 0, and Z ⊂ X a proper closed subset. Let n = dim X, and again assume we know the weak resolution theorem for varieties of dimension n − 1.

First a few reduction steps:

1. Show, as in 4.1 that we may assume X projective and normal, and Z the support of a Cartier divisor.

2. Show that there is a finite projection X → ℙ^{n}.

3. Let P → ℙ^{n} be the blowup at a closed point. Show that

P ≃ ℙ_{ℙ^{n−1}}(O_{ℙ^{n−1}} ⊕ O_{ℙ^{n−1}}(1)).

Denote by E the exceptional divisor of P → ℙ^{n}.

4. By blowing up a general point on ℙ^{n}, and blowing up X at the points above, show that we may assume we have a finite morphism f : X → P, which is étale along E, such that the image of Z is disjoint from E, and maps finitely to ℙ^{n−1}.

5. By the Nagata - Zariski purity theorem, note that the branch locus of X → P is a divisor B, disjoint from E, mapping finitely to ℙ^{n−1}.

We replace Z by Z ∪ f^{−1}B.

7.2 Vector bundles.

The next steps are aimed at replacing P by another ℙ^{1}-bundle Q → ℙ^{n−1}, such that the branch locus in Q of X → Q becomes simpler. Let Y be any variety, F a rank-2 vector bundle on Y, P = ℙ_{Y} (E). Let E ⊂ P be a divisor which is a section of π : P → Y and let D ⊂ P be another effective divisor disjoint from E. Denote by O_{P} (1) the tautological bundle, and by d the relative degree of D over Y.

1. Consider the exact sequence

0 → I_{D}(d) → O_{P} (d) → O_{P} (d)|_{D} → 0.

Use this to show that there is an invertible sheaf L_{D} on Y such that I_{D}(d) ≃ π^{∗}L_{D}.

2. If D_{1}, D_{2} ⊂ P are any two disjoint divisors finite of degree d over Y, show that there is an embedding of vector bundles L_{D_{1}} ⊕ L_{D_{2}} ⊂ sym^{d}F inducing a surjection π^{∗}(L_{D_{1}} ⊕ L_{D_{2}}) → O_{P} (d).

3. Assume the characteristic is 0. Consider the case D_{1} = dE, D_{2} = D. show that the resulting morphism P → ℙ_{Y} (L_{D_{1}} ⊕ L_{D_{2}}) = P′ maps E to a section E′ and D to a disjoint section; and its branch locus is of the form (d − 1)E + D′ where D′ has degree d − 1 over Y and is disjoint from E′.

4. Continue by induction to show that there is a ℙ^{1} bundle Q → Y and a morphism g : P → Q over Y, such that the image of D and the branch locus of g form a union of sections of π_{Q} : Q → Y.

7.3 Conclusion of the proof.

Back to our theorem, where Y = ℙ^{n−1}. Composing with the morphism f : X → P, we obtain that the image g(f(Z)) ⊂ Q is the image D_{1} of a section s_{1} : ℙ^{n−1} → Q of Q → ℙ^{n−1} and the banch locus of g ∘ f is the union of images D_{i} sections s_{i} : ℙ^{n−1} → Q as well. Denote Δ = π_{Q}(∪_{i≠j}D_{i} ∩ D_{j}).

The following steps use moduli theory; however it has been shown (in the preprint version of [12], and in Paranjape’s exposition [52]) that this can be easily circumvented.

1. Use the stable reduction argument to show that there is a modification Y′ → Y, and a modification Q′ → Q ×_{Y} Y′ such that Q′ → Y is a family of nodal curves of genus 0, and the sections lift to disjoint sections s′_{i} : Y′ → Q′. We replace Y by Y′, Δ by its inverse image, etc.

2. Use induction on the dimension to replace Y by a nonsingular variety such that Δ becomes a strict divisor of normal crossings.

3. Use either toroidal geometry, or Section 4.11, to replace Q by a nonsingular variety, such that the inverse image of D is a strict normal crossings divisor.

4. Let X̃ be the normalization of Q in the function field of X. Use Abhyankar’s lemma to show that X̃ has a toroidal structure, such that the inverse image Z̃ of Z is a toroidal divisor.

5. Conclude that there is a weak resolution of singularities r : X′ → X such that r^{−1}Z is a strict divisor of normal crossings.

8 Weak resolution of singularities II

The weak resolution argument according to Abramovich - de Jong starts very much like de Jong’s theorem: a projection X → P is produced, and a Galois alteration P_{1} → P over which one has stable reduction X_{1} → P_{1}, equivariant under the Galois group G, is produced. Induction on the dimension for P allows one to assume that X_{1} and P_{1} are toroidal, and the Galois action on P_{1} is toroidal as well. The only point left is to make the group action on X_{1} toroidal, so that the quotient should be toroidal, and therefore admit toroidal resolution.

Let us go through the steps. Let X be a variety over an algbraically closed field of characteristic 0, and let Z ⊂ X be a Zariski-closed subset. We want to find a nonsingular, quasi-projective variety X′ and a modification f : X′ → X such that f^{−1}Z is a divisor with simple normal crossings.

8.1 Reduction steps.

Exercise 8.1. Show that it is enough to prove the result when X is projective and normal, and Z a Cartier divisor.

Exercise 8.2. Reduce to the case where there is a projection X → P, such that the generic fiber is a smooth curve.

Exercise 8.3. ∗Using the trick of enlarging Z and stable reduction, show that there is a diagram as follows:

\begin{aligned}X_{1} & → & X \\ ↓ & & ↓ \\ P_{1} & → & P \end{aligned}

such that P_{1} → P is an alteration, X_{1} → X̃_{P_{1}} is birational, and X_{1} → P_{1} has section s_{i} : P_{1} → X_{1} making it a family of stable pointed curves, and the image of these sections in X contains Z.

Exercise 8.4. Show that you can make P_{1} → P a Galois alteration. Call the Galois group G. Show, using the uniqueness of stable reduction3.14, that the action of G on P_{1} lifts to an action on X_{1}, which permutes the sections s_{i}.

You can replace X by X_{1}/G and P by P_{1}/G

Exercise 8.5. Use induction on the dimension to reduce to the case where:

1. P is nonsingular, with a normal crossings divisor Δ;

2. The branch locus of P_{1} → P is contained in δ;

3. The locus where X_{1} → P is not smooth is contained in δ.

Exercise 8.6. Show that in this case P_{1} → P is a toroidal morphism, G acts toroidally on X, and X_{1} → P_{1} is a toroidal morphism as well.

The only point left is to make the action of G on X_{1} toroidal - if it were, then X would be toroidal and we could easily resolve its singularities.

Looking locally, the question boils down to the following situation:

Let T_{0} ⊂ X_{0} be an affine torus embedding, X_{0} = Spec R. Let G ⊂ T_{0} be a finite subgroup of T_{0}, let p_{0} ∈ X_{0} be a fixed point of the action of G, and let ψ_{u} be a character of G. Consider the torus embedding of T = T_{0} × Spec k[u, u^{−1}] into X = X_{0} × Spec k[u], and let G act on u via the character ψ_{u}. Write p = (p_{0}, 0) ∈ X and write D = (X_{0} T_{0})×Spec k[u]. We wish to find a canonical blowup X_{1} → X, such that if U ⊂ X_{1} is the inverse image of T_{0}, then it is a toroidal embedding, and the group G acts toroidally.

8.2 The ideal

Let M ⊂ R[u] be the set of monomials. For each t ∈ M let χ_{t} be the associated character of T, and let let ψ_{t} : G → k^{∗} be the restriction of χ to G. Define M_{u} = {t|ψ_{t} = ψ_{u}}, the set of mononials on which G acts as it acts on u. Define I_{G} = ⟨M_{u}⟩, the ideal generated by M_{u}.

Exercise 8.7 (canonicity). Show that if X′_{0}, T′_{0}, G′, p′_{0} and ψ′_{u} is a second set of such data, and if we have an isomorphism of formal completions

φ : X̂_{p} →^{∼} X̂′_{p′},

which induces isomorphisms G ≅ G′ and D̂_{p} ≅ D̂′_{p′}, then φ pulls back I_{G} to the ideal I_{G′}.

Exercise 8.8 (gluing property). If q_{0} is any point of X_{0} and if G_{q} ⊂ G is the stabilizer of q = (q_{0}, 0) in G, then the stalk of I_{G} at q is the same as the stalk of I_{G_{q}} at q.

Exercise 8.9. Show that I_{G} is generated by u and a finite number of monomials t_{1}, …, t_{m} in M_{u} ∩ R.

Exercise 8.10. Let X′ = B_{I_{G}}(X) be the blowup. Let X′_{u} be the chart with coordinates u, t_{j}/u. Show that the action of G on X′_{u} is toroidal.

Exercise 8.11. Ley X′_{i} be the chart on X′ with coordinates t_{i}, v = u/t_{i}, s_{j} = t_{j}/t_{i}. Show that G acts trivially on v, and that X′_{i} = Spec R′_{i}[v] where R′_{i} is generated over R by s_{j}.

Exercise 8.12. Let X_{1} be the normalization of X′. Show that if U ⊂ X_{1} is the inverse image of T_{0}, then it is a toroidal embedding, and the group G acts toroidally.

9 Intersection multiplicities

Intersection theory has a long history, and certainly we are not going to say much about it here. One aspect is, that it is not so easy to have a good definition for intersection multiplicities. Remark, exercise: let C ⊂ ℙ_{k}^{2} be a plane algebraic curve, P ∈ C a closed point at which C is regular, and D = Z(F) ⊂ ℙ^{2} a plane curve given by a homogeneous polynomial F; suppose F is not identical zero on a neighborhood of P in C (i.e. no component of D contains the component of C containing P). Show that the following two definitions of the intersection multiplicity i(C, D; P) of C and D at P are equivalent:

• the dimension of the k-vector space

O_{C,P} ⊗_{O_{ℙ^{2},P}} O_{D,P} ,

• the value of the valuation v = v_{C,P} defined by the discrete valuation ring O_{C,P} computed on the function on C given by F,

see [HAG], Exercise (5.4) on page 36, and Remark (7.8.1) on page 54.

Consider two varieties V, W ⊂ ℙ^{n} which have an isolated point of intersection at P ∈ V ∩ W. One could try to define the intersection of V and W at P as the length of

O_{V,P} ⊗_{O_{ℙ^{n},P}} O_{W,P} .

Analogous situations of intersections of arbitrary schemes in some regular ambient scheme can be considered.

Exercise 9.1. (See Gröbner [23], 144.10/11, also see [64], [60], [10], see [HAG], I.7): a) Let C ⊂ ℙ^{3} be the space curve with parameterization

(x_{1} : x_{2} : x_{3} : x_{4}) = (t^{4} : t^{3}·s : t·s^{3} : s^{4})

(we work over some field K). Show that the prime ideal given by this curve equals

j := (T_{1}^{2}T_{3}−T_{2}^{3} , T_{1}T_{4}−T_{2}T_{3} , T_{1}T_{3}^{2}−T_{2}^{2}T_{4} , T_{2}T_{4}^{2}−T_{3}^{3}) ⊂ K[T_{1}, T_{2}, T_{3}, T_{4}].

b) Consider C as a curve embedded in ℙ^{4}: choose the hyper plane ℙ^{3} ≅ Z(T_{0}) = H, and we get C ⊂ H ⊂ ℙ^{4}. Let P := (x_{0} = 1 : 0 : 0 : 0 : 0) ∈ ℙ^{4}. Define V ⊂ ℙ^{4} as the cone with vertex P over the curve C ⊂ ℙ^{4}, i.e. V is defined by the ideal

J := K[T_{0}, T_{1}, T_{2}, T_{3}, T_{4}] · j, V = Z(J).

Note that the dimension of V equals two, that the degree of V ⊂ ℙ^{4} equals four.

c) Let W be the 2-plane given by

I := (T_{1}, T_{4}) ⊂ K[T_{0}, T_{1}, T_{2}, T_{3}, T_{4}], W := Z(I).

Note that P ∈ W. Remark that (set-theoretically):

W ∩ V = {P }

(use the geometric situation, or give an algebraic computation). We like to have a Bézout type of theorem for the situation W ∩V ⊂ ℙ^{4}, however:

d) Define

M := O_{W,P} , A := O_{ℙ^{4},P} , N := O_{V,P} ,

and compute

dim_{K} (M ⊗_{A} N)

(surprise: this is not equal to four).

e) Compute

dim_{K} (Tor_{i}^{A}(M, N)) , ∀i

(either using, or reproving χ_{A}(M, N) = 4, for notation see below).

Hence we see that just the length of the appropriate tensor product does not define necessarily the correct concept. Serre proposed in 1957/58 to define the intersection multiplicity as the alternating sum of the lengths of the Tor_{i} (note that Tor_{0} = ⊗), i.e. by

χ_{A}(M, N) := ∑_{i≥0} length_{A} Tor_{i}^{A}(M, N)

(we follow notation of [64], also see [10], 6.1, see [60]), here A is a regular local ring, and M and N are A-modules such that M ⊗_{A} N has finite length. In equal characteristic this is the right geometric concept (i.e. satisfies Bézout’s theorem, coincides with previously defined intersection multiplicities etc.).

The following theorem was Conjectured by Serre, proved by Gabber (using de Jong’s alteration result), and written up by Berthelot (in [10], 6.1):

Theorem 9.2. Let the characteristic of A be equal to zero. Suppose p ∈ 𝔪^{2}, hence its residue field A/𝔪 has characteristic p > 0. Then:

χ_{A}(M, N) ≥ 0.

Part II

Moduli of curves

10 Introduction to moduli of curves

It is an important feature of algebraic geometry, that the set of all objects (e.g. smooth projective curves) of the same a fixed geometric nature (e.g. genus) often has the structure of an algebraic variety itself. Such a space is a “moduli space”, which gives a good algebraic meaning to the problem of “classification”. It is fair to say that this “self referential” nature of algebraic geometry is one of the main reasons for the depth of the subject - it is impossible to overestimate its importance.

The first instances of this phenomenon to be discovered were those of embedded variety: the projective space as a parameter space for lines in a vector space; Grassmannians parametrizing vector subspaces of arbitrary dimension; the projective space (of dimension (d^{2} + 3d)/2) parametrizing all plane curves of degree d, and so on. The case of abstract varieties, such as smooth curves, had to await for some technical advances, although already Riemann knew that algebraic curves of genus g “vary in 3g − 3 parameters”.

Historically moduli spaces of curves, or of curves with points on them, were constructed with more or less ad hoc methods. Moduli spaces for 3 or 4 points on rational curves have been known for ages, using the so called “cross ratio” (see exercise 10.13 below). For genus 1, the modular function j was used (see exercise 10.17). The case of genus 2 was already quite difficult to acheive by algebraic methods [30]. For years, moduli spaces for higher genus were only known to exist using Teichmüller theory.

One problem which took years to solve was, that no good understanding of “what moduli spaces really are” was available. Then Grothendieck introduced the notion of “representable functor”, describing the best possible meaning for moduli spaces. This had a great success with the development of Hilbert schemes. For a while one hoped that nature would be as ideal as expected (see Grothendieck hopeful Conjecture 8.1 in [24] 212-18, and its retraction in the Additif of [24] 221-28). But it was soon seen that in general moduli functors are not representable, or as we say now, some moduli functors do not give rise to “fine moduli spaces” due to existence of automorphisms. Finally, Mumford pinned down the compromise notion of a “coarse moduli scheme”, which enables us to have a good insight in various aspects of moduli theory. This is what we shall try to describe here. It should be said that since then, other good approaches were developed, by way of “enlarging the category of schemes” to include some “moduli objects”, called stacks. For details see [17], Section 4. We will not pursue this direction here.

In this section we gather some basic definitions on functors of moduli for curves. In the next sections we discuss existence theorems for moduli spaces of curves, and for complete moduli spaces with extra structure carrying a “tautological family”. Section 14 is devoted to some further questions, examples and facts, not needed for the methods of alterations, but in order to give a more complete picture of this topic.

10.1 The functor of points and representability

To any scheme M one naturally associates a contravariant functor

F_{M} : {Schemes} → {Sets}

via

X ↦ Mor(X, M).

This is known as the functor of points of M, see [Red Book], II §6.

We say that a contravariant functor F : {Schemes} → {Sets} is representable by a scheme M, if it is isomorphic to F_{M}, i.e. there is a functorial isomorphism

ξ : F(−) ⟶^{∼} Mor_{S}(−, M).

Remark 10.1. Strictly speaking, it is the pair (M, ξ), consisting of the object M and the isomorphism ξ, which represent F. But it has become customary to say “M represents F”, suppressing ξ.

Already in the early Bourbaki literature one finds this notion in the disguise of a “universal property”. The question of representability of functos can also be posed in categories other than the category of schemes.

Exercise 10.2. 1. Fix an integer N, and let V be a vector space of dimension N + 1 over ℂ. Consider the functor F_{ℂ} that associates to a scheme T over ℂ, the set {L ⊂ T × V } of all line sub-bundles of the trivial vector bundle T × V. Show that F_{ℂ} is represented by ℙ_{ℂ}^{N}.

2. Let F be the functor that associates to any scheme T (over ℤ) the set {L ⊂ O_{T}^{N+1}} of all locally free subsheaves of rank 1 of the trivial sheaf O_{T}^{N+1} having locally free quotient. Show that F is represented by the projective scheme ℙ^{N}.

3. In general, show that the grassmannian scheme Grass(n, r) represents the functor of locally free subsheaves of rank r of the trivial free sheaf O_{T}^{n} of rank n having locally free quotients.

Exercise 10.3. Fix integers N and d, and let G be the functor that associates to a scheme T the set {X ⊂ ℙ_{T}^{N}} of all flat families of hypersurfaces of degree d in projective N-space over T. Show that G is represented (over ℤ) by a projective space ℙ^{M−1}, where M = \binom{N+d}{d} is the dimension of the space of homogeneous polynomials of degree d in N + 1 variables.

Exercise 10.4. Show that the Hilbert scheme H_{ℙ^{N} ,P (T)} represents the “Hilbert” functor, that associates to a scheme T, the set of all subschemes X ⊂ ℙ_{T}^{N}, which are flat over T and such that the geometric fibers have Hilbert polynomial P (T).

10.2 Moduli functors and fine moduli schemes.

Suppose a contravariant functor F has the nature of a moduli functor, namely, it assigns to a scheme S the set {C → S}/ ≅ of isomorphism classes of certain families of objects over S. As a guiding example, let us fix an integer g, with g ∈ ℤ_{≥0}, and define the moduli functor for smooth curves:

M_{g}(S) = {isom. classes of families of curves of genus g over S}.

A morphism T → S defines (by pulling back families) a map of sets in the opposite direction: M_{g}(T) ← M_{g}(S), and we have obtained a contravariant functor.

Assume the functor F were represented by a scheme M. Then we would call M a fine moduli scheme for this functor F, and the object

C → M corresponding to the identity id ∈ F(M) = Mor_{S}(M, M)

would be called a universal family.

Remark 10.5. Note that in the exercise above on the Hilbert scheme, we can view it as a fine moduli scheme, if we agree that “families up to isomorphism” means “up to isomorphisms as subfamilies of the fixed ℙ_{T}^{N}, namely up to equality.

It is a fact of life that for every g ≥ 0 the functor M_{g} is not representable. We will explain later why this is true in general, but for the moment let us consider the easiest case:

Exercise 10.6. Let us say that C is a “curve of genus 0”, if it is an algebraic curve defined over a field K, and over some extension of K ⊂ L it is isomorphic with ℙ_{L}^{1}. In other words: C is geometrically irreducible, reduced, it is complete and of genus equal to zero.

1. Let K be a field. Show there exist an extension K ⊂ K′, and two curves of genus 0 over K′ which are not isomorphic.

2. For every algebraically closed field k, the set M_{0}(k) consists of one element, M_{0}(k) = {ℙ_{k}^{1}}.

3. Show that the moduli functor M_{0} is not representable.

10.3 Historical interlude.

The first case of a highly nontrivial algebraic construction of a moduli space of curves, appeared in Igusa’s work [30]. This is a construction of a “moduli scheme for non-singular curves of genus two in all characteristics”, which would now be denoted by M_{2} → Spec(ℤ). This happened almost concurrently with Grothendieck’s study of representability of functors. But notice that, when Samuel discussed these beautiful results by Igusa in Séminarie Bourbaki (see [61]), his very first comment was:

“Signalons ausitôt que le travail d’ Igusa ne résoud pas pour les courbes de genre 2, le “problème des modules” tel qu’il a été posé par Grothendieck à diverses reprises dans ce Séminaire.”

It really seemed that Nature was working against algebraic geometrers, refusing to provide us with these fine moduli schemes...

The truth is, Nature does provide us with a replacement. Indeed, not much later, Mumford (see [GIT], 5.2) discovered how to follow nature’s dictations and come to a good working definition, requiring that the scheme gives geometrically what you want, and does it in the best possible way.

10.4 Coarse moduli schemes

Here is the definition:

Definition 10.7. A scheme M and a morphism of funcors

φ : F → Mor_{S}(−, M)

is called a coarse moduli scheme for F if:

1. for every algebraically closed field k the map

φ(k) : F (Spec(k)) → Mor_{S}(Spec(k), M) = M(k)

is bijective, and

2. for any scheme N and any morphism ψ : F (−) → Mor_{S}(−, N) there is a unique χ : N → M factoring ψ.

And here is its first triumphant success:

Theorem 10.8 (Mumford). Suppose g ≥ 2. The functor M_{g} of smooth curves of genus g admits a quasi-projective coarse moduli scheme.

See [GIT], or [17], Corol. 7.14. We will denote the coarse moduli scheme of M_{g} by M_{g} → Spec ℤ.

We note some properties:

• As we mentioned before, for every g ≥ 2 the functor M_{g} is not representable: there does not exist a universal family of curves over M_{g} which can give an isomorphism between M_{g} and M_{g}.

• For every g ≥ 2 and for any field K, the variety (M_{g})_{K} = M_{g} ×_{Spec ℤ} Spec K is not complete. A fortiori, the morphism M_{g} → Spec(ℤ) is not proper.

• At least for the sake of de Jong’s theorem, we need a moduli space of curves with points on them.

The first problem is solved by introducing a finite covering M → M_{g} admitting a tautological family, namely a family realizing the morphism M → M_{g} as its moduli morphism. The nicest way of doing this is by introducing a new moduli functor, of smooth curves “enriched” with a finite amount of “extra structure”, which does admit a fine moduli scheme. See Section 13.

In order to “compactify” these spaces, the notion of stable curves was invented. Historically, the influential paper [17] by Deligne and Mumford seems to be one of the first printed versions in which the concept of stability, especially in the case of algebraic curves is explained and used. In [45] we see that already in 1964 Mumford was trying to find the appropriate notions assuring good compactifications. In [GIT], page 228, Mumford attributes the notion of a stable curve to unpublished joint work with Alan Mayer.

As it turns out, the third problem was solved almost concurrently with the second.

First, the moduli space of smooth pointed curves:

Theorem 10.9. Let g ∈ ℤ_{≥0}, and n ∈ ℤ_{≥0} such that 2g − 2 + n > 0. Consider the functor M_{g,n} of isomorphism classes of families of stable smooth n-pointed curves of genus g. This functor admits a quasi-projective coarse moduli scheme.

We will denote this moduli space by M_{g,n} → Spec(ℤ).

Remark 10.10. 1. Note that this includes the previous theorem.

2. For g = 0 and n = 3 this space is proper over Spec(ℤ). However in all other cases in the theorem M_{g,n} is not proper. In many cases it will not represent the functor (see Section 14 for a further discussion), in other words, in general this is not a fine moduli scheme.

3. It is important to note that these spaces exist over Spec(ℤ), which is useful for arithmetical applications.

4. The litterature poses difficulties in choosing notations. In [GIT] the subscript n denotes a level structure, but in [39] it indicates the number of marked points. We have chosen to indicate the markings as lower index, using n, and the level structure as upper index, using (m).

Finally the moduli space of stable pointed curves:

Theorem 10.11 (Knudsen and Mumford). Let g ∈ ℤ_{≥0}, and n ∈ ℤ_{≥0} such that 2g − 2 + n > 0. Consider the functor \overline{M_{g,n}} of isomorphism classes of families of stable n-pointed curves of genus g. This functor admits a projective coarse moduli scheme.

See [39], part II, Theorem 2.7 and part III, Theorem 6.1, or [22], Theorem 2.0.2. We will denote this moduli scheme by \overline{M_{g,n}} → Spec(ℤ).

The following exercise should give you an idea why the moduli space \overline{M_{0,n}} is complete. This is discussed in further detail in the next section.

Exercise 10.12. Let K be a field, and let R ⊂ K be a discrete valuation ring having K as field of fractions. Consider the projective line ℙ^{1} over K and suppose n ≥ 3, let P_{1}, ⋯, P_{n} ∈ ℙ^{1}(K) are distinct points. Write P = {P_{1}, ⋯, P_{n}}. Construct a stable n-pointed curve (C, P) → Spec(R) extending (C, P). (You will need to blow up closed points over the special fiber where the Zariski closures of P_{i} meet. Then you may need to blow down some components! See [39].)

Exercise 10.13. 1. Let K be a field. Given three distinct finite points P_{1}, P_{2} and P_{3} on ℙ_{k}^{1} consider the cross ratio

λ(P_{1}, P_{2}, P_{3}, z) = \frac{(z − P_{1})(P_{2} − P_{3})}{(z − P_{3})(P_{2} − P_{1})}.

Show that, as a function of z, the cross ratio is an automorphism of ℙ^{1} carrying P_{1}, P_{2}, P_{3} to 0, 1, and ∞, respectively. Show that this automorphism is the unique one with this property. Check that this definition can be extended to the case where one of the points is ∞.

2. Using the cross ratio λ defined above, describe M_{0,3}.

3. Show that M_{0,3} is a fine moduli scheme by exhibiting a universal family over it!

4. Show that M_{0,3} = \overline{M_{0,3}}.

5. Use the cross ratio to give an explicit description of M_{0,4}. Show that it is a fine moduli scheme by explicitly constructing a universal family.

6. Use the above (possibly together with the previous exercise) to describe M_{0,4} ⊂ \overline{M_{0,4}}.

7. Show that \overline{M_{0,4}} is a fine moduli scheme, and give explicit descriptions of the universal family.

8. Show that the universal family over \overline{M_{0,4}} is canonically isomorphic to \overline{M_{0,5}}.

Exercise 10.14. Give an alternative description of \overline{M_{0,4}} as follows: consider the projective space ℙ of dimension 5 parametrizing conics in ℙ^{2}. Choose four points in general position in ℙ^{2} (for instance (1 : 0 : 0), (0 : 1 : 0), (0 : 0 : 1), (1 : 1 : 1) will do). Let M ⊂ ℙ be the subscheme parametrizing conics passing through these four points. Show that M = \overline{M_{0,4}} and the universal family of conics is a universal family for \overline{M_{0,4}}.

Exercise 10.15.

1. Show that M_{0,n} exists and is a fine moduli scheme (you may exhibit it as an open subscheme of (ℙ^{1})^{n}).

2. Show that, assuming \overline{M_{0,n}} is a fine moduli scheme, then there is a canonical morphism \overline{M_{0,n+1}} → \overline{M_{0,n}} which exhibits \overline{M_{0,n+1}} as the universal family over \overline{M_{0,n}}.

3. ∗ Show that for every n ≥ 3, the scheme \overline{M_{0,n}} is a fine moduli scheme. (You may want to use Knudsen’s stabilization technique.)

Remark 10.16. For every n ≥ 3, let (C, P) be a stable n-pointed rational curve. Then Aut(C, P) = {id}. You do not need to know this in the previous exercise, but it “explains” why the result should be true.

Exercise 10.17. Let k be a field of characteristic ≠ 2 and let (E, O) be an elliptic curve, namely a projective, smooth and connected curve E of genus 1 with a k-rational point O on it.

1. Considering the linear series of O_{E}(2O), show that E can be exhibited as a branch covering of ℙ^{1} of degree 2.

2. Show that the branch divisor B on ℙ^{1} is reduced and has degree 4.

3. If k is algebraically closed, show that E is determined up to isomorphism by the divisor B.

4. Conclude that M_{1,1} is isomorphic to the quotient of M_{0,4} by the action of the symmetric group S_{4}, permuting the four points.

5. Assume further that char k ≠ 3, so that every elliptic curve can be written in affine coordinates as y^{2} = x^{3} + ax + b. Show that

j(E) = 1728\frac{4a^{3}}{4a^{3} + 27b^{2}}

is an invariant characterizing the k̄-isomorphism class of E, exhibiting M_{1,1} = 𝔸^{1}.

11 Stable reduction and completeness of moduli spaces

11.1 General theory

In order to understand the reason why \overline{M_{g,n}} is projective, let us recall the following:

Theorem 11.1 (The valuative criterion for properness). A morphism

f : X → Y

of finite type is proper, if and only if the following holds:

Let R be a discrete valuation ring, and let S := Spec(R) be the corresponding “germ of a non-singular curve”, with generic point η. Let φ : S → Y and let ψ_{η} : η → X be a lifting:

\begin{matrix}η & →^{ψ_{η}} & X \\ ↓ & & ↓ \\ S & →^{φ} & Y. \end{matrix}

Then there is an extension ψ : Γ → X, lifting φ:

\begin{matrix}η & →^{ψ_{η}} & X \\ ↓ & ↗^{ψ} & ↓ \\ S & →^{φ} & Y. \end{matrix}

See [HAG], II, Theorem (4.7) for a precise formulation.

Let us translate this to our moduli scheme. Keeping in mind the relationship between the functor \overline{M_{g,n}} and the space \overline{M_{g,n}}, one might hope that every family of stable pointed curves over η as in the theorem above might extend to R. This is not the case, as we shall see later. However, a weaker result, sometimes called “the weak valuative criterion for properness”, does hold for the functor \overline{M_{g,n}}, and it does imply the valuative criterion for \overline{M_{g,n}}. The first case to consider is when the generic fiber is smooth and n = 0. This is the content of the following result, the Stable Reduction Theorem for a one parameter family of curves:

Theorem 11.2. Let S = Spec(R) be the spectrum of a discrete valuation ring R, η ∈ S the generic point, corresponding with the field of fractions K of R. Let C_{η} → η be a smooth stable curve of genus g > 1. There exists a finite extension of discrete valuation rings R ↪ R_{1}, with S_{1} = Spec R_{1} and generic point η_{1}, and an extension

C_{η_{1}} ↪ C_{1}
↓  ↓
{η_{1}} ↪ S_{1},

such that C_{1} → S_{1} is a family of stable curves.

Proofs of this theorem, using different methods, may be found in various references. One proof which works in pure characteristic 0 is relatively simple. As the reader will notice, none of the general proofs is easy or elementary.

Most proofs of this theorem use resolution of singularities of 2-dimensional schemes (Abhyankar).

Exercise 11.3. Suppose R is of pure characteristic 0. Let s ∈ Spec R be the closed point.

1. Show that there exists an extension π : C → S such that π is proper and flat, C is nonsingular, and C_{s} ⊂ C is a normal crossings divisor.

2. Let x ∈ C_{s} be a singular point. After passing to the algebraic closure of the field of constant, let x̄ ∈ Ĉk̄ be the completion. Show that one can find local parameters u, v at x̄ and t at s̄ ∈ Ŝk̄, and positive integers k_{x}, l_{x}, such that t = u^{k_{x}}v^{l_{x}}.

3. Let S_{1} → S be a finite cover obtained by extracting the n-th root of a uniformizer, where n is divisible by all the non-zero k_{x}, l_{x} given above. Let C′_{1} be the normalization of C ×_{S} S_{1}. Show that the special fiber is reduced and nodal.

4. Show that the minimal model C_{1} of C′_{1} → S_{1} is stable.

We list some approaches for positive and mixed characteristic:

Artin-Winters. This proof can be found in [7]. A precise and nice description and analysis of the proof is given by Raynaud, see [58].

In this proof one attaches an numerical invariant to a given genus, and one proves that by choosing a prime number q larger than this invariant, and not equal to the residue characteristic, and by extending the field of definition of a curve of that genus such that all q-torsion point on the jacobian are rational over the extension, then one acquires stable reduction. The proof consists of a careful numerical analysis of the possible intersection matrices of components of degenerating curves. The proof does not rely on a lot of theory, but is quite subtle.

Grothendieck, Deligne-Mumford. This proof can be found in [17], Theorem (2.4) and Corollary (2.7).

In this proof one shows that a curve has stable reduction if and oly if its jacobian has stable reduction. Then one shows following Grothendieck that eigenvalues of algebraic ℓ-adic monodromy are roots of unity (see [65], Appendix). Moreover, again following Grothendieck one shows that these eigenvalues are all equal to one iff the abelian variety in question has stable reduction. The advantage of this proof is that it has a more conceptual basis. The big disadvantage is that it relies on the theory of Néron models, whose foundations are quite difficult.

Hilbert schemes and GIT - Gieseker. See [22], Chapter 2, Proposition(0.0.2). He says on the first page of the introduction: “...we use results of Chapter 1 to give an indirect proof that the n-canonical embedding of a stable curve is stable if n ≥ 10, and to construct the projective moduli space for stable curves. As corollaries, we obtain proofs of the stable reduction theorem for curves, and of the irreducibility for smooth curves.” The proof uses Geometric Invariant Theory to prove directly that \overline{M_{g}} exists and is projective, and then one can easily derive the theorem. This proof does not use resolution of singularities for surfaces in any explicit manner.

Remark 11.4. This theorem is an instance of the semistable reduction problem. In [10], 1.3, the definition of semistable reduction, over a one-dimensional base, and arbitrary fiber dimension, is recalled. As we have seen above, it is true that if the relative dimension is one, stable reduction, hence semistable reduction, exists over a one-dimensional base. For higher relative dimension an analogous result holds in pure characteristic zero - see [38]. The general case is an important open problem, which seems difficult.

Once Theorem 11.2 is known, it is easy to generalize it. The pointed case can be easily proven using Knudsen’s stabilization technique:

Exercise 11.5. Let R be a discrete valuation ring, with field of fractions K, suppose (C, p_{1}, …, p_{n}) is a smooth, stable n-pointed curve of genus g > 1 defined over K. There exists a finite extension R ⊂ R_{1} of valuation rings, with K_{1} the field of fractions of R_{1}, such that C_{1} = C ⊗ K_{1} extends to a stable n-pointed curve C_{1} → Spec(R_{1}).

The case of genus zero follows from Exercise 10.13. We will discuss the case of genus 1 in Section 11.2 below.

We can also consider the case when the generic fiber is not necessarily smooth:

Exercise 11.6. Let R be a discrete valuation ring, with field of fractions K, suppose C is a stable curve defined over K. There exists a finite extension R ⊂ R_{1} of valuation rings, with K_{1} the field of fractions of R_{1}, such that C_{1} = C ⊗ K_{1} extends to a stable curve C_{1} → Spec(R_{1}). [Below we formulate a generalization to stable pointed curves of this.]

We now give a full generalization of (11.2):

Exercise 11.7. Let S be the spectrum of a discrete valuation ring, η ∈ S the generic point. Let (C, P) → {η} be a stable n-pointed curve of genus. I.e., C is a complete, nodal curve defined over a field K, and P := {P_{1}, ⋯, P_{n}} are distinct closed points P_{j} ∈ C(K), with such that (C, P) is stable n-pointed over K.

Then there exists a finite extension of discrete valuation rings S ↪ S_{1}, with generic point η_{1}, and an extension

(C_{η_{1}}, P) ↪ (C, P)
↓  ↓
Spec(K_{1}) ↪ S_{1}

such that (C, P) → S_{1} is a family of stable n-pointed curves.

This is the “weak valuative criterion for properness” of the functor \overline{M_{g,n}}.

Remark 11.8. Here is a hint about a technical detail which can be used in solving the previous exercises, “The normalization of a stable n-pointed curve”: Suppose given a stable n-pointed curve (C, P) over a field K, with P = {P_{1}, ⋯, P_{n}}. There exists a finite extension K ⊂ L, a finite disjoint union (D, Q) of stable pointed curves, and a morphism (“the normalization”) φ : (D, Q) → (C, P)_{L} = (C, P)⊗_{K} L such that: D = ∐ D^{(t)}, let the singular points of C_{L} be: R_{j} ∈ C(L), with 1 ≤ j ≤ d, moreover Q = {Q_{1}, ⋯, Q_{n}} ∪ {S_{j}, T_{j} | 1 ≤ j ≤ d}, for every irreducible component of C_{L} there is a unique component of D mapping birationally onto it, the morphism φ is an isomorphism outside Sing(C_{L}), the markings Q_{i} corresponds with the markings P_{i} of the pointed curve (C, P), and the markings {S_{j}, T_{j}} are precisely the points mapping to R_{j}.

You need to show this choice can be made, and show it is unique in case K = k is an algebraically closed field.

Corollary 11.9. Let g ∈ ℤ_{≥0}, and n ∈ ℤ_{≥0} such that 2g − 2 + n > 0. The coarse moduli scheme π : \overline{M_{g,n}} → Spec(ℤ) is proper over Spec(ℤ).

Proof: We use the valuative criterion for properness setting X = \overline{M_{g,n}} and Y = Spec(ℤ). Suppose R is a discrete valuation ring, with field of fractions K, and suppose given

Spec K →^{ψ_{K}} X
↓  ↓
Spec R →^{φ} Y.

By the definition of a coarse moduli scheme, there is a finite extension K ⊂ K′ such that the point ψ_{K}(Spec(K)) ∈ X corresponds to a stable pointed curve (C, P) over K′. By the stable reduction theorem there is a finite extension K′ ⊂ K_{1} such that (C, P) ×_{Spec K′} Spec K_{1} extends to a stable pointed curve; this defines a morphism τ : Spec(R_{1}) → X, “extending” φ and ψ_{K}. It factors over Spec(R), because R = K ∩ R_{1}. This shows that the condition for the valuative criterion holds in our situation, hence that π : \overline{M_{g,n}} → Spec(ℤ) is proper.

11.2 Stable reduction for elliptic curves.

In these exercises we illustrate the concept of stable reduction by studying the case of elliptic curves. In this case examples are easy to give because in many cases we can choose plane models (Weierstrass equations). These exercises can be used at motivation for more abstract methods which apply for higher genus. You can do the exercises by explicit methods and calculations.

For details on elliptic curves, Weierstrass equations, the j-function, and related issues, see [67], Chapters III and VII.

A non-singular one-pointed curve of genus one is called an elliptic curve. In other words: an elliptic curve is an algebraic curve E defined over a field, absolutely irreducible, non-singular, of genus one, with a marked point P ∈ E(K). Morphisms are supposed to respect the marked point.

The following exercise is an easy exercise using the theorem of Riemann-Roch.

Exercise 11.10. Show the following are equivalent:

1. (E, P) is an elliptic curve over K.

2. E ⊂ ℙ_{K}^{2} is a plane, nonsingular cubic curve, with a marked point P ∈ E(K).

3. (E, P) is an abelian variety of dimension one over K.

Definition 11.11. Let R = R_{v} be a discrete valuation ring, with K = fract(R) its field of fractions, and k = R_{v}/m_{v} the residue class field.

1. An elliptic curve E defined over K is said to have good reduction (at the given valuation) if there exists a smooth proper morphism E → Spec(R) with generic fibre isomorphic to E → Spec(K). If E does not have good reduction, we say that it has bad reduction.

2. We say E has stable reduction at v if either it has good reduction, or there exists a nodal E → Spec(R) with generic fibre isomorphic to E → Spec(K).

Definition 11.12. We say that E has potentially good reduction, if there exists a finite extension K ⊂ L, where B is the integral closure of R in L, and w a valuation over v, such that E ⊗ L has good reduction at w.

We define potentially stable reduction analogousely.

Here are some exercises to warm up:

Exercise 11.13. Suppose R = k[T], with char(k) ≠ 2, ≠ 3, and let E over K = k(T) be given by the equation Y^{2} = X^{3} + T^{6}. Show that E has good reduction at the valuation given by v(T) = 1.

Exercise 11.14. Suppose R = k[S], with char(k) ≠ 2, ≠ 3, and let E over K = k(S) be given by the equation Y^{2} = X^{3} + S. Show that E has bad reduction at the valuation v given by v(S) = 1.

Exercise 11.15. Suppose R = k[S], with char(k) ≠ 2, ≠ 3, and let E over K = k(S) be given by the equation Y^{2} = X^{3} + S. Show that E has potentially good reduction at the valuation v given by v(S) = 1.

Suppose that E is given over K by a Weierstrass equation with coefficients in R (see [67], III). Such an equation defines an affine plane curve E ⊂ 𝔸_{B}^{2} over Spec(R) = B, and it is easy to see that the curve E_{0} := E ⊗_{R} k is irreducible and has at most one singular point. The curve E is obtained by adding the point at infinity to E_{0}. Suppose the Weierstrass equation is minimal at v. If this singular point is a cusp, we say that this reduction is of additive type, if it is a node we say that this reduction is of multiplicative type, or we say in this case the reduction is stable.

Exercise 11.16. Show that the notion of “good reduction” as defined earlier is equivalent by saying there is Weierstrass equation defining good reduction. Show that a reduction of multiplicative type is a stable reduction.

A reduction given by a minimal Weierstrass equation of additive type is bad reduction which is non-stable; non-stable bad reduction is sometimes called cuspidal reduction.

Exercise 11.17. Suppose R = k[T], with char k ≠ 2, and let E over K = k(T) be given by the equation Y^{2} = X^{2} · (X − 1). Show that any model of this curve given by a Weierstrass equation has stable reduction at the valuation given by v(T) = 1. Show that this curve does not have potentially good reduction.

Exercise 11.18. Let R be a discrete valuation ring, with residue characteristic ≠ 2, and fraction field K. Let E be an elliptic curve over K.

1. Show that after a suitable extension of R, the curve E admits a minimal Weierstrass equation of the form

y^{2} = x(x − 1)(x − λ)

for some λ ∈ R.

2. Conclude that this curve has potentially stable reduction.

Exercise 11.19. ∗ Let R_{v} be a DVR, with residue characteristic ≠ 3. Suppose E is an elliptic curve over K given by a Weierstrass equation E = V (F) ⊂ ℙ_{K}^{2} such that all flex points of E have coordinates in K.

1. Show that this curve admits a plane equation (not a Weierstrass equation!) over K of the form

λX^{3} + Y^{3} + Z^{3} = 3μXY Z,

for some (λ : μ) ∈ ℙ_{R_{v}}^{1}.

2. Show that E ⊗_{K} L has stable reduction at v.

3. Show that E has potentially stable reduction.

Conclusion. Every elliptic curve over a field K with a discrete valuation has potentially stable reduction at that valuation.

This is a special case of 11.7, the stable reduction theorem for stable curves of arbitrary genus.

Exercise 11.20. ∗

1. Let R_{v} be a DVR, and E an elliptic curve of the fraction field K. Show that E has potentially good reduction at R_{v} if and only if j(E) ∈ R_{v}.

2. Can you formulate (and prove?) the same result for curves of arbitrary genus?

3. If E is an elliptic curve over a field K, and End_{K}(E) ≠ ℤ, then E has potentially good reduction at every place of K.

11.3 Remarks about monodromy

Let C be a non-singular curve over a field K, and let v be a discrete valuation of K. Consider properties of good reduction, bad reduction at v, and so on. We have quoted that C has stable reduction at v iff J := Jac(C) has stable reduction, see [17], Proposition (2.3).

However note that it may happen that J has good reduction, and C has bad reduction; this is the case if the special fibre C_{0} of the minimal model of C at v has a generalized jacobian J_{0} = Jac(C_{0}) which is an abelian variety. Such a curve C_{0} is called a curve of “compact type”, or a “nice curve” (and sometimes called a “good curve”, but we do not like that terminology, because a curve reducing to a “good curve” may not have good reduction...). In this case the special fibre C_{0} is a tree of non-singular curves, i.e. every irreducible component is non-singular, and in the dual graph of C_{0} there are no cycles. The easiest example is: a join of two non-singular curves, each of genus at least one, meeting transversally at one singular point. For example a curve of genus two degenerating to a transversal crossing of two curves of genus one is the easiest example. Here is another example: take ℙ^{1} with three marked points, and attach three elliptic tails via normal crossings at the markings, arriving at a nice curve of genus three

Monodromy (action of the local fundamental group of the base on cohomology) decides about the reduction of an abelian variety being bad or good, see [65], Theorem 1 on page 493. In the analytic context one can take the local fundamental group of a punctured disc acting on cohomology; in all cases one considers the inertia-Galois group of v acting on ℓ-adic cohomology, where ℓ is a prime number not equal to the residue characteristic of v.

Note that algebraic monodromy has eigenvalues which are roots of unity. This was proved by Landman, Steenbrink, Brieskorn in various settings, and we find a proof by Grothendieck in the appendix of [65]. For a sketch of that proof, see [48], for further references, see [51].

Algebraic monodromy is trivial iff X = J has good reduction, iff C has compact type reduction (which may be either good reduction or bad but “nice” as explained above).

The algebraic monodromy is unipotent (all eigenvalues are equal to one) if and only if X has stable reduction, if and only if C has stable reduction.

But, how can we distinguish for curves the difference between good reduction and bad compact type reduction? As we have seen, this is not possible via algebraic monodromy on cohomology. But, in a beautiful paper, [8] we find a method which for curves unravels these subtle differences for curves: the local fundamental group of the base acts via outer automorphisms on the fundamental group of the generic fibre (again, here one can work in the analytic-topological context, or in the ℓ-adic algebraic context). This action is trivial iff C has good reduction.

12 Construction of moduli spaces.

Early constructions of the moduli spaces of smooth curves M_{g} included a complex - analytic constructions via Teichmüller theory and via the construction of moduli of abelian variety using locally symmetric spaces. These constructions are not algebraic in nature and therefore cannot be generalized to positive or mixed characteristics.

A first algebraic approach, which is still commonly used today, was given by Mumford using his Geometric Invariant Theory [GIT]. We will sketch one version of this approach, due to Gieseker, which automatically gives also the moduli spaces of stable curves \overline{M_{g}}. There is another approach, due to Artin and Kollár [40], which circumvents the use of Geometric Invariant Theory. Nowadays both approaches work over ℤ.

How does one start? It is evident that if we want to parametrize all stable curves of a certain genus, we had better have some family of curves in which all these curves appear. We know of two general approaches for that. One method uses parameter spaces for curves embedded in projective space, such as Hilbert schemes (or Chow varieties). We will follow this approach. The other approach, due to Artin [6], uses versal deformation spaces. It works in greater generality but involves a number of technicality which we would rather avoid here.

It is easy to see that for any stable curve C of genus g > 1, and any ν ≥ 3, the ν-canonical series H^{0}(C, ω_{C}^{ν}) gives an embedding of C as a curve of degree d := ν(2g − 2) in a projective space of dimension N := ν(2g − 2) − g. Thus the Hilbert scheme H_{ℙ^{N} ,P (T)} (over ℤ!) parametrizing subschemes of ℙ^{N} with Hilbert polynomial P (T) := dT + 1 − g carries a universal family C_{ℙ^{N} ,P (T)} → H_{ℙ^{N} ,P (T)} in which each stable curve of genus g appears at least once.

There are two problems with this family:

1. Each curve appears more than once in the family. Indeed, the embedding of the curve C in ℙ^{N} involves two choices: a choice of a line bundle of degree d, and a choice of a basis for the linear series. And of course the curves could also be embedded in a projective subspace using a subseries.

2. There are many curves in ℙ^{N} with Hilbert polynomial P (T) which are far from stable.

Since a nodal curve can only deform into nodal curves, it is easy to see that there is an open subset H_{st} ⊂ H_{ℙ^{N} ,P (T)} which parametrizes stable curves, embedded by a complete linear system in ℙ^{N}. Denote the restriction of the universal family to H_{st} by π : C_{st} → H_{st}. Considering the locus in H_{st} where R^{1}π_{∗}(O(1) ⊗ ω_{C_{st}→H_{st}}^{ν}) jumps in dimension, we immediately see that there is a closed subscheme H_{g} ⊂ H_{st} parametrizing stable curves embedded by a complete ν-canonical series. The restiction of the universal family will be denoted C_{g} → H_{g}.

There is a natural action of the projective linear group P GL(N +1) on H_{g} via changing coordinates on ℙ^{N}. It is easy to see that the “ambiguity” for choosing the embedding of a curve C in the latter universal family is fully accounted for by the action of this group. In other words, stable curves correspond in a oneto one manner with P GL(N + 1) orbits in H_{g}. Thus, at least set theoretically, \overline{M_{g}} = H_{g}/P GL(N + 1).

12.1 Geometric Invariant Theory and Gieseker’s approach.

We arrived at the following questions:

1. Does the quotient H_{g}/P GL(N + 1) exist as a scheme?

2. Can we show that it is projective?

3. Does it satisfy the requirements of a coarse moduli scheme?

Geometric Invariant Theory is a method which allows one to approach the first two questions simultaneousely. The third question then becomes an easy gluing exercise.

The general situation is as follows: Let X ⊂ ℙ^{n} be a quasi-projective scheme and suppose G is an algebraic group acting on ℙ^{n} and stabilizing X. One wants to know whether or not a quotient X/G exists as a scheme and whether or not it is projective.

A natural approach is to look for a space of invariant sections of some line bundle. Thus assume that the action of G on ℙ^{n} lifts to O_{ℙ^{n}}(l). Then it also lifts to any power O_{ℙ^{n}}(l · m), and we can look at the ring of invariants R := ⊕(O_{ℙ^{n}}(l · m)^{G}. We have a natural rational map q : ℙ^{n} −−→ R. We would like to know whether or not this map is well defined along X, and what the image is like.

First, an easy observation. For any point x ∈ X, the map q is well defined at x if and only if there exists a nonconstant invariant f ∈ R such that f(x) ≠ 0.

We want to check whether q is a quotient map at x. To go any further, we need to assume that the group G is reductive. Assuming that G is reductive, then the question whether map q is a quotient map at a neighborhood of q can be translated to a question about the closure \overline{Gx} of the orbit of x: one needs to check that for any point y ∈ \overline{Gx} Gx there is an invariant f ∈ R which vanishes at y but not at x. A point x is called GIT-stable if it satisfies this condition.

Mumford’s numerical criterion for stability (see [GIT]) gives a way to check GIT-stability in some situations.

Let us consider our situation. The scheme H_{g} is quasi projective - from its construction one sees that it naturally sits inside a grassmannian, which has a Plc̈ker embedding in some ℙ^{n}. It is easy to see that the action of P GL(N + 1) extends to ℙ^{n}, and lifts to some line bundle O_{ℙ^{n}}(k). Applying this criterion systematically, Gieseker verified in [22] that

1. If a point x ∈ H_{ℙ^{N} ,P (T)} corresponds to a scheme which is not a stable curve, or to a curve which is not embedded by a complete linear series, then every nonconstant invariant vanishes at x.

2. If a point x ∈ H_{ℙ^{N} ,P (T)} corresponds to a stable curve embedded by the complete ν-canonical linear series, then x is GIT-stable.

Using the two statements, and the fact that G is reductive, it is not difficult to realize that

1. the map H_{g} → ℙ^{n} is a quotient map, and

2. the image of H_{g} is projective.

This proves the existence and projectivity of \overline{M_{g}}.

12.2 Existence of \overline{M_{g,n}}.

There is no known analogue of Gieseker’s result for stable pointed curves. It is not difficult to construct a Hilbert-type scheme for stable pointed curves, with a reductive group action, and such that the quotient is set-theoretically \overline{M_{g,n}}. But in order to tell that the quotient is isomorphic to \overline{M_{g,n}} as a scheme, we first need to construct \overline{M_{g,n}} in some other way.

But there is a very useful trick, which reduces the construction of \overline{M_{g,n}} to the existence of \overline{M_{g}} for some larger value of g. We give the reduction over a field, but it works similarly over ℤ.

Fix n irreducible stable curves C_{i} of genus g_{i} > g, all nonisomorphic to each other, and fix a rational point x_{i} ∈ C_{i}. For any stable n-pointed curve (C, p_{1}, …, p_{n}) of genus g, we can construct a stable curve C′ of genus g′ = g + ∑ g_{i} as follows: C′ = (∪C_{i}) ∪ C, where we glue together C and C_{i} by identifying p_{i} with x_{i}.

Clearly this construction gives a set theoretic embedding \overline{M_{g,n}} → \overline{M_{g′}}. The image set is easily seen to be a scheme, and by working the construction in a family it is easy to see that it is a coarse moduli scheme.

13 Existence of tautological families.

For almost any application of moduli spaces of curves including the alteration theorem, it is necessary to know that there exists a family C → M over a scheme M such that the associated morphism to the moduli space is finite and surjective. Such a family is called a tautological family. Various authors have devised general methods of showing this, but for the moduli spaces of curves there is a “very nice” way to find such a cover, using level structures. The case of the modui space \overline{M_{g}} of stable (unpointed) curves is discussed in detail in [21]. In this section we describe how this can be generalized for sable pointed curves as well. We rely throughout on the treatment in [21]. In Section 13.4 we outline another way to construct tautological families, which works in greater generality.

13.1 Hilbert schemes and level structures.

Fix:

• an integer g ∈ ℤ_{≥0} (the genus),

• an integer n ∈ ℤ_{≥0} (the number of marked points),

• such that 2g − 2 + n > 0,

• and an integer m ∈ ℤ_{≥1} (the level).

• Fix an integer ν ∈ ℤ_{≥5}, which will be used to study ν-canonical embeddings of curves into a projective space.

Remark 13.1. If n = 0 or m = 1 these data will be omitted from the notation, e.g. M_{g,0} = M_{g}. If g = 0, the level structure is irrelevant, M_{0,n}^{(m)} = M_{0,n}.

Let C be a curve whose jacobian is an abelian variety. By a level-m structure on C we mean a symplectic level structure as explained in [21]. If a level-m structure is considered we assume that all schemes, varieties are over a base on which m is invertible, i.e. are schemes over Spec(ℤ[1/m]).

Recall that there is a Hilbert scheme H_{P} parametrizing curves C ⊂ ℙ^{N}, where N = ν·(2g − 2 + n) − g, with Hilbert polynomial P (t) = ν·(2g − 2 + n) · t − g + 1. We want to find a scheme parametrizing pointed curves - this is done in a standard way as follows. Observe that there is a closed subscheme H_{P,n} ⊂ H_{P} × (ℙ^{N})^{n} parametrising pairs (C, (p_{1}, …, p_{n})) where p_{i} ∈ C. There is an open subscheme H_{st} ⊂ H_{P,n} where the curves are nodal, the points are distinct and smooth points on the curves, and the pairs (C, (p_{1}, …, p_{n})) are stable. Last, there is a closed subscheme H_{g,n} ⊂ H_{st} where the embedding line bundle of C ⊂ ℙ^{N} is isomorphic to (ω_{C}(p_{1} + … + p_{n}))^{ν}.

Over H_{g,n} there is a universal family C_{g,n} → H_{g,n} with sections s_{i} : H_{g,n} → C_{g,n} of stable pointed curves, embedded in ℙ^{N} by the chosen line budle. The linear group PGL = P GL(N) acts on C_{g,n} → H_{g,n} equivariantly, and \overline{M_{g,n}} = H_{g,n}/PGL is the quotient.

Note that there is an open subet H_{st}^{0} ⊂ H_{g,n} parametrizing smooth stable pointed curves.

13.2 Moduli with level structure.

Theorem 13.2. For m ≥ 3, and 2g − 2 + n > 0, there exists a fine moduli scheme M_{g,n}^{(m)} for smooth stable n-pointed curves with level-m structure. In particular there exists a universal curve with level structure over M_{g,n}^{(m)}. This moduli scheme is smooth over Spec(ℤ[1/m]).

Note in particular that M_{g,n}^{(m)} is a normal scheme, and that M_{g,n}^{(m)} → M_{g,n} is a Galois cover with Galois group Sp(2g, ℤ/m).

We use the notation S_{m} := Spec(ℤ[1/m]).

Definition 13.3. Let g ∈ ℤ_{≥1}. Fix n ∈ ℤ_{≥0}, with 2g − 2 + n > 0. For any m ∈ ℤ_{≥3}, the scheme

\overline{M_{g,n}^{(m)}} ⟶ S_{m}

is defined as the normalization of \overline{M_{g,n}}[1/m] = \overline{M_{g,n}} ×_{ℤ} S_{m} in M_{g,n}^{(m)}.

For simplicity of notation in this section, we write M = \overline{M_{g,n}^{(m)}} and M^{0} = M_{g,n}^{(m)} ⊂ M.

Theorem 13.4. Fix g, n, and m as above. Suppose m ≥ 3. There exist a stable n-pointed curve (C, P) → M, and a level-m-structure α on C^{0} := C|_{M^{0}} such that

(C, P) → M is tautological for M → \overline{M_{g,n}},

and such that

(C^{0}, P^{0}, α) → M^{0}

represents the functor M_{g,n}^{(m)}.

We give an argument for 13.2 and 13.4 following the line of [21]. This is a kind of “boot-strap” argument, which uses the idea that once one quotient space exists, many others follow. We also sketch another argument which reduces the problem to the case of [21].

There is a relative jacobian scheme J(C_{g,n}^{0}) → H_{g,n}^{0}. This is an abelian scheme, so we can look at its group-subscheme of m-torsion points. Taking a symplectic rigidification of this group scheme we arrive at H_{g,n}^{(m),0} - the Hilbert scheme of smooth stable n-pointed curves with symplectic level-m structure, embedded in projective space as above.

The action of PGL on H_{g,n}^{0} clearly lifts to H_{g,n}^{(m),0}. This immediately implies that

M_{g,n}^{(m)} = H_{g,n} / PGL

exists, since it is finite over M_{g,n}. By Serre’s lemma this action has no fixed points, and it also lifts to C_{g,n}^{(m),0} = C_{g,n} ×_{H_{g,n}} H_{g,n}^{(m),0}. This means that the quotient PGL\C_{g,n}^{(m),0} → M_{g,n}^{(m)} is a universal family of smooth stable pointed curves with level structure.

This proves Theorem 13.2.

The normalization of H_{g,n} in H_{g,n}^{(m),0} will be denoted by H_{g,n}^{(m)}. The argument of [21], (2.6) works word for word, and shows that PGL still acts without fixed points on H_{g,n}^{(m)}. This gives the existence of the quotient

\overline{M_{g,n}^{(m)}} = H_{g,n}^{(m)} / PGL.

Again the universal family over H_{g,n}^{(m)} descends to a family over \overline{M_{g,n}^{(m)}}, this extends the universal family over M_{g,n}^{(m)}, and clearly it is tautological. This proves Theorem 13.4.

13.3 Proof by reduction to the unpointed case

Starting from \overline{M_{g}^{(m)}} and its tautological family we can construct \overline{M_{g,n}^{(m)}} and its tautological family by induction on the number of points n in the manner described below.

Denote by D → \overline{M_{g,n}^{(m)}} the tautological family. It is easy to see that in fact

D = \overline{M_{g,n+1}^{(m)}}. So D ×\overline{M_{g,n}^{(m)}} D → \overline{M_{g,n+1}^{(m)}} is a family of stable n-pointed curves

with level structure, but with an additional section given by the diagonal. Using the stabilization process as described in [39] (see Section 3.7 above) one blows

this scheme up, to obtain the tautological family over \overline{M_{g,n+1}^{(m)}} as desired.

Remark 13.5. The moduli space M_{g,n}^{(m)} is smooth over S_{m} for m ≥ 3; this follows from Serre’s lemma and deformation theory. However, the moduli space \overline{M_{g,n}^{(m)}} is singular if g > 2; Serre’s lemma holds also in this situation, but the space is not the coarse (or fine) moduli space of a moduli functor whose deformation spaces coincide with the deformations of stable curves. For more explanation, see [44] or [21].

The argument above works for g > 1 when \overline{M_{g}^{(m)}} exists. For rational curves these theorem are relatively easy, and known, since the moduli spaces are fine moduli spaces in genus 0. For elliptic curves these theorems are known by the theory of modular curves.

13.4 Artin’s approach via slicing

A general approach for constructing tautological families over finite covers of coarse moduli space was developed by Artin (see description in [40]). Here we present a version of this approach adapted to stable pointed curves.

Step 1: slicing Consider the locally closed subset of the Hilbert scheme H_{g,n} discussed above. It carries a universal family of stable pointed curves C_{g,n} → H_{g,n} suitably embedded in a projective space. This family induces a natural morphism H_{g,n} → \overline{M_{g,n}}. The fibers coincide with the G-orbits associated to the embedded curves, where G = PGL.

Fix a point x ∈ H_{g,n}. By repeatedly taking hyperplane sections, we can find a locally closed subscheme V_{x} ⊂ H_{g,n} such that

1. Gx ∩ V_{x} ≠ ∅;

2. If x′ ∈ H_{g,n} and Gx′ ∩ V_{x} ≠ ∅, then there exists a neighborhood x′ ∈ U such that for any y ∈ U we have that Gy ∩ V ≠ ∅; and

3. for any y ∈ H_{g,n} we have that V_{x} ∩ Gy consists of finitely many closed points.

These V_{x} are “multi-sections” of the map H_{g,n} → \overline{M_{g,n}} in a neighborhood of Gx. The essential point is that all orbits in H_{g,n} are of the same dimension.

Using the Noetherian property, we can choose finitely many of these, say V_{1}, …, V_{l}, such that every orbit meets at least one of them.

Step 2: normalization. Let K be the join of the function firlds of V_{i} over \overline{M_{g,n}}. Let V be the normalization of \overline{M_{g,n}} in the Galois closure of K. The scheme V admits many rational maps to the V_{i}. It is not hard to see that for every point v ∈ V at least one of these maps is well defined at v! Pulling back the families on V_{i}, we see that V is covered by open sets, each of which carries a family of stable pointed curves, compatible with the given morphism V → \overline{M_{g,n}}.

Step 3: Gluing. Now we can use lemma 3.19 inductively. We obtain a finite surjective M → V over which the families glue together to a family C → M such that the associated moduli morphism is the composition M → V → \overline{M_{g,n}}. Since V is finite over \overline{M_{g,n}}, this forms a tautological family.

Remark 13.6. It is not hard to construct a tautological as above without using the existence of \overline{M_{g,n}}! One can use this to construct the moduli space “from scratch” as a proper algebraic space, which is roughly speaking a quotient of a scheme by a finite equivalence relation. Kollár in [40] has shown how to use this to prove, without GIT, that \overline{M_{g,n}} is projective.

14 Moduli, Automorphisms, and families

This section will not be needed in the proofs above. The central theme here is the relationship between automorphisms, coarseness of moduli, and the existence of families. The main principle which will emerge is:

A moduli space M is a fine moduli space

⇕

objects parametrized by M have no nontrivial automorphisms

⇕

M carries a unique tautological family.

We also touch on the issue of singularities of moduli spaces.

For rational curves, and n ≥ 3, the moduli schemes M_{0,n} and \overline{M_{0,n}} exist, these are smooth over Spec(ℤ), these are fine moduli schemes, i.e. they carry a universal family.

However, the moduli space M_{1,1} and the moduli spaces M_{g} for g > 1 are not fine for the related moduli functor.

Exercise 14.1 (Deuring). Let K be a field, let x ∈ K. Then there exists an elliptic curve E defined over k with j(E) = x. [ Suppose char(K) ≠ 2, ≠ 3, suppose E is is given over K

by the equation Y^{2} = X^{3} + AX + B, with 4A^{3} + 27B^{2} ≠ 0. Then define

j(E) := 1728·4·A^{3}/(4A^{3} + 27B^{2}).

For the definition of the j-invariant, see [67].]

This can partly be made more precise as follows:

Exercise 14.2. Consider M_{0,1} ≅ 𝔸_{ℤ}^{1}, and remove the sections j = 0 and j = 1728:

U := 𝔸_{ℤ}^{1} {0, 1728}_{ℤ}.

There exists a tautological curve

E → U.

1. This cannot be extended over any of the deleted points.

2. This family is not at all unique.

Exercise 14.3. Consider U := ℂ {0, 1728}. Show: up to isomorphisms there exist exactly 4 tautological curves (stable, one pointed smooth curves of genus 1 with j invariant different from 0 and 1728) over this moduli space. In particular, show that for the ground field K = ℚ there are infinitely many tautological curves over the moduli space 𝔸_{ℚ}^{1} {0, 1728}. Characterize them all.

Now we have seen the difference between a universal curve and a tautological curve: the moduli problem for elliptic curves with geometrically no non-trivial automorphisms admits a coarse moduli scheme; over that scheme there is a tautological curve, but the scheme is not a fine moduli scheme (not every family is a pull-back from one chosen tautological curve). Here is another example:

Definition 14.4. A curve C → S is called a hyperelliptic curve if it is smooth, of relative genus g with g ≥ 2, and if there exists an involution ι ∈ Aut(C/S) such that the quotient C/ < ι >→ S is a smooth family of rational curves.

Remark 14.5. Elliptic curves and rational curves are not called “hyperelliptic”, but sometimes the terminology “quasi-hyperelliptic” is used for curves having an involution with rational quotient.

Theorem 14.6. Consider the moduli space Hip_{g} of hyperelliptic curves of genus g ≥ 2 (even over ℂ). If g is even there does not exist a curve defined over teh function field ℂ(Hip_{g}) having as moduli point the generic point of Hip_{g}.

(See Shimura [66], Theorem 3.)

In different terminology: For no open dense subset U ⊂ H_{g} does there exist a tautological curve when g is even.

There does exist a open dense subset U ⊂ H_{g} and a tautological curve C_{U} when g is odd.

Corollary 14.7. No dense open subset in M_{2} or in M_{2} ⊗ K carries a tautological curve.

Exercise 14.8. Choose g ∈ ℤ_{>2}, and consider nonsingular curves of genus g.

1. Show that there exists such a curve which has no nontrivial automorphisms.

2. (variant:) Show that a general curve of genus > 2 has no nontrivial automorphisms.

Remark 14.9. There is a morphism M_{g,n+1} → M_{g,n} (”forgetting the last marking”). Sometimes this is called the “universal curve over M_{g,n}, but we think in general this terminology is not justified in all cases possible.

Theorem 14.10. Let U ⊂ M_{g} with g ≥ 3 fixed, be the set of points corresponding with curves which have geometrically no non-trivial automorphisms. This set is dense and open. Let M_{U} be the corresponding moduli functor. This functor is representable.

In other terminology: there does exist a (unique) universal curve C_{U} → U for the moduli problem of curves of genus g ≥ 3 with geometrically no non-trivial automorphisms.

In particular: Let K be a field, g ∈ ℤ_{≥3}, and η be the generic point of M_{g} ⊗ K. There exists an algebraic curve defined over K(η) having η as moduli point. However the universal family as indicated above over U ⊂ M_{g} does not extend to any smooth family of curves over M_{g}.

Exercise 14.11. Formulate and prove a generalization of previous theorems to the case of stable pointed curves.

Exercise 14.12. Let n > 2g+2 and let (C, P_{1}, ⋯, P_{n}) be any stable n-pointed curve of genus g. Suppose that C is regular (and hence irreducible). Show that

Aut((C, P_{1}, ⋯, P_{n})) = {1}

(if you want, assume that char(k) = 0).

Exercise 14.13. Let g ∈ ℤ_{≥1} and 2 − 2g < n ≤ 2g + 2 and 0 ≤ n. Show that M_{g,n} is a coarse, but not a fine moduli space.

Exercise 14.14. Choose g ∈ ℤ_{≥0}, and let n > 2g + 2. Show that M_{g,n} is a fine moduli space. Show that the universal curve over M_{g,n} is not smooth if n ≥ 2.

Exercise 14.15. Consider all stable n-pointed curves of genus g. Suppose that

2g − 2 + n ≥ 3.

1. Show that there exists such a curve which has no nontrivial automorphisms.

2. (variant:) Show that a general curve as above has no nontrivial automorphisms.

Exercise 14.16. Choose some g (e.g. g = 3), choose a very large integer n (e.g. n = 1997), and construct a stable n-pointed curve of genus g which has a non-trivial group of automorphisms.

Variant: Let 2g − 2 + n ≥ 2; show that there exist stable n-pointed curves of genus g ≥ 3 in codimension two in the moduli space with non-trivial groups of automorphisms.

Exercise 14.17. Let g ∈ ℤ_{≥1}, and n > 2 − 2g and n ≥ 0. Show that \overline{M_{g,n}} is not a fine moduli space.

Choose 2g − 2 + n > 0, choose m ≥ 1 and let M be one of the following spaces: M_{g,n}^{(m)}, or \overline{M_{g,n}} (all these spaces are defined by a moduli functor). Let x ∈ M(k) be a geometric point, and let X_{0} := (C, P, α) be the corresponding object over k (if C is non-smooth there is no level structure, the genus of C is g, we have P = ∅ if n = 0, we have α = id if m = 1). Let D = Def(X_{0}) be the universal deformation space; i.e. consider Λ = k if char(k) = 0, and Λ = W_{∞}(k) in case of positive characteristic, consider all local artin Λ-algebras, and consider the object prorepresenting all deformations of X_{0} over such algebras (see [62]). This universal deformation object exists, and it is formally smooth over Λ on 3g − 3 + n variables; in case n = 0 this can be found in [17], page 81, the case of pointed curves follows along the same lines; in case m > 1, we have required that m is invertible in k, finite, flat group schemes of m-power order on such bases are étale, and deformations of level structures are unique by EGA IV^{4}, 18.1. Let G := Aut(X_{0}). Note that G is a finite group (because we work with stable curves). Note that G acts in a natural way on D = Def(X_{0}) by “transport of structure”.

Theorem 14.18. In the case described, the formal completion of M at x is canonically isomorphic with the quotient

Def(X_{0}) / G ⟶^{∼} M_{x}^{∧}.

This is well-known, e.g. see [27], §1.

Exercise 14.19. (Rauch, Popp): Let g ∈ ℤ_{≥4}, and let A ⊂ M_{g} be an irreducible component of the set of all points corresponding with curves with non-trivial automorphisms. Show that the codimension of A ⊂ M_{g} is ≥ 2. (In positive characteristic this is also correct, but you might need some extra insight to prove also those cases.)

Remark 14.20. Stable rational pointed curves have no non-trivial automorphisms. For elliptic curves there are curves with more than 2 automorphisms in codimension one. For curves of genus two we find a description of all curves with “many automorphisms” in [30]. Note that hyperelliptic curves of genus three are in codimension one.

Exercise 14.21. Show that non-hyperelliptic curves of genus three with nontrivial automorphisms are in codimension at least two.

Exercise 14.22. (Rauch [57], Popp [56]): Let g ∈ ℤ_{≥4}, and let [C] = x ∈ M_{g} be a geometric point. Show that x is a singular point on M_{g} iff Aut ≠ {id}. [You might like to use: [5], Coroll. 3.6 on page 95: A quasi-finite local homomorphism of regular local rings having the same dimension is flat. Also you might like to use purity of branch locus: a ramified flat covering is ramified in codimension one.]

Remark 14.23. For singularities of M_{2} see [30]. Show that for genus three nonhyperelliptic points are singular iff there are non-trivial automorphisms, e.g. see [49]. For singularities of moduli schemes of abelian varieties, see [50].

Remark 14.24. As we have seen in [21], the moduli schemes \overline{M_{g}^{(m)}} have singularities for all g ≥ 3 and m ≥ 3 (these spaces cannot be handled with the methods just discussed, these spaces are not given by “an obvious” moduli functor !). As Looijenga, see [41], in characteristic zero, and Pikaart and De Jong, see [53] showed, there exist a finite map M → M_{g} with M regular (using nonabelian level structures) (it is even true that M is smooth over ℚ, or smooth over ℤ[1/r] for some natural number r > 1).

Summary about

M_{g,n}^{(m)} ↪ \overline{M_{g,n}^{(m)}} ⟶ Spec(ℤ[1/m])

for

g ∈ ℤ_{≥0}, n ∈ ℤ_{≥0}, m ∈ ℤ_{≥1}, with 2g − 2 + n > 0,

M_{g,n}^{(m)} and \overline{M_{g,n}} exist as coarse moduli schemes, we have constructed \overline{M_{g,n}^{(m)}}. We have seen:

• For g ≥ 2 the coarse moduli scheme M_{g} → S = Spec(ℤ) exists. These are not fine moduli spaces. They do not carry a tautological family. For every g this is singular.

• For g ≥ 2 the coarse moduli scheme \overline{M_{g}} → S = Spec(ℤ) exists. These are not fine moduli spaces. They do not carry a tautological family. They are singular.

• A dense open set in M_{1,1} carries a tautological family, and it is not universal.

• No dense open set in M_{2} carries a tautological family.

• For g ≥ 3 a dense open set in M_{g,n} carries a universal family.

• For n ≥ 3 the moduli spaces M_{0,n} ⊂ \overline{M_{0,n}} exist, they are fine moduli spaces, they are smooth over S = Spec(ℤ).

• For 2g−2+n > 0, and m ≥ 0 the moduli spaces M_{g,n} → S, and \overline{M_{g,n}} → S and M_{g}^{(m)} → S_{m} exist, they coarsely represent a moduli functor. For n > 2g + 2 the moduli space M_{g,n} is fine, and smooth over Spec(ℤ) (but the universal family is not smooth for n > 1). For m ≥ 3 the space M_{g}^{(m)} is fine and smooth over S_{m}.

• For 2g − 2 + n > 0, and m ≥ 0 there is a moduli space, and a tautological family, with properties as in 13.4. For g ≥ 3 the morphism \overline{M_{g,n}^{(m)}} → S_{m} is not smooth.

References

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Dan Abramovich Department of Mathematics Boston University 111 Cummington Street Boston, MA 02215 USA

abrmovic@math.bu.edu
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Frans Oort

Mathematisch Instituut University of Utrecht Budapestlaan 6 NL - 3508 TA UTRECHT The Netherlands

                                            oort@math.uu.nl
                                            http://www.math.uu.nl/staff/oort.html