ALTERATIONS AND RESOLUTION OF SINGULARITIES

Course material for “Mountains and Singularities” working week on resolution of singularities September 7 - 14, 1997, Obergurgl, Tirol, Austria

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

             abrmovic@math.bu.edu

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

                                              oort@math.ruu.nl

January 29, 1998 INFORMAL NOTES

1. Introduction

On July 26, 1995, at the University of California, Santa Cruz, a young Dutch mathematician by the name A.J. de Jong made a revolution in the study of the arithmetic, geometry and cohomology theory of varieties in positive or mixed characteristic (his results were already announced in the Arbeitstagung, Bonn, June 1995). 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.

These pages include informal preliminary material for a course to be given by the authors during the Working Week on Resolution of Singularities, September 7-14, 1997 in Obergurgl, Tirol, Austria. In the course, we intend to explain de Jong’s results in detail, with proofs as complete as possible, and give some applications, both arithmetic and in characteristic zero. Hopefully, the participants will come to share some of the excitement we felt on that beautiful July day in Santa Cruz.

2. Prerequisites

We are trying to keep the prerequisetes for the course at minimum, but some basics are necessary.

The participants are expected to have a good understanding of basic algebraic geometry. The standard text is Hartshorne’s book [HAG]: Chapter II is essential, as well as most of Chapter III (though you can take some parts of it on faith). A good alternative to parts of [HAG], is Mumford’s Red Book [M.RedBook]. This book does a great job at building one’s intuition for the subject.

De Jong’s results build on the theory of moduli of curves. We hope the participants will take some time before the course to read about the subject and build some intuition for it. The functor of points of a scheme is introduced in [M.RedBook], II§6. Lecture 21 in Harris’s book [Harris] is good bed-time reading about parameter spaces and moduli spaces. Chapter IV in Eisenbud - Harris [E-H] is a bit more detailed, and will also give an understanding of the functor of points. Finally, Mumford’s lectures [M.C+J] give an excellent introduction to the theory of moduli of curves, from a more analytic point of view. The standard reference for moduli spaces is [M.GIT].

For the applications in characteristic 0, we will use the theory of toroidal embeddings. The participants should be familiar with toric varieties, as in the first two chapters in Fulton’s book [F], or in [KKMS], Chapter I, §1-2. This theory will also be discussed in the parallel course given by B. Teissier. If one wants to become familiar with toroidal embeddings in advance, one could look in [KKMS], Chapter II, §1-2.

Finally, if you would like to learn about the structure of de Jong’s proofs, you may want to read the introduction of de Jong’s paper [dJ], or Berthelot’s Bourbaki seminar [BerBour]

2.1. To summarize, please read and study: here is the list of prerequisites we will take for granted in our course:

1. Basic algebraic geometry: [HAG] Chapters I and II (we will also use III). Alternatively, or in addition, read [M.RedBook], Chapters I and II;

2. Moduli of curves: in addition to the above, we suggest [M.GIT], pp. 96-104, and Th. 5.11 (p. 115); please have a look at [D-M], say pp. 75-96; please see [Knu], II;

3. Toroidal geometry: the first two chapters in Fulton’s book [F], or [KKMS], Chapter I, §1-2.

2.2. Supplementary reading: if you want, and if you have time and energy it might be good to have a look at topics like:

• Please read [dJ], it is nice if you already have tasted the flavour of the subject,

• for ℓ-adic representations, see [S-T], see [SGA], pp.326-354, or see [O];

• for toroidal embeddings, [KKMS], Chapter II, §1-2.

2.3. This pamphlet. One important fact you can conclude from the discussion above is that the material in this pamphlet is not a prerequisite for the course. If you can understand all the material in this pamphlet, do all the exercises, and see how the proofs should go, the course will not be able to compete with the attractions of the Alps around. What we put in this pamphlet are three types of data:

1. a few definitions which we will use;

2. statements and discussion of the main results we wish to discuss, without proofs; and 3. a preliminary form for some sets of exercises.

We hope that you will be able to read the definitions and get some idea about the statement of results. You may also try your hands at some of the exercises. All of this may help you get an idea about what the course is about. It may also help you formulate for yourself some questions which will hopefully be answered during the course.

The exercises are certainly not in their final form. We intend to rewrite them and add many more, of various types - some will help you prepare, some will be done during the course, and some will be intended for further investigation.

Part 1. Results and tools

3. Some preliminaries and generalities on varieties

3.1. Definition. To fix notation, 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 (or for one) algebraically closed field k_{1} containing k. A variety then is defined by glueing a finite number of affine varieties in a separated way. For more information, please see [M.RedBook], I.5 definition 1 (p. 35) and I.6 definition 2 (p. 52).

3.2. Remark. This definition differs slightly from that in [dJ]. De Jong requires the algebraic scheme to be integral, 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.

We now introduce some important types of operations on varieties:

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

3.4. Definition (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 [dJ], 2.20.

3.5. Remark. A modification is a birational alteration. We adopt the terminology: we say we have a resolution of singularities if we achieve a regular variety by a modification. If we can only do this by an alteration, we shall not use the word “resolution”. We agree that this is the right way to avoid possible confusion.

3.6. Exercise. 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.

3.7. Exercise. 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.

3.8. Remark. 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.

To pick up the “important part” of the pullback by an alteration, we do the following:

3.9. Definition (de Jong). Let Y → Z be an alteration, f : X → Z a morphism of varieties. The strict transform f′ : X̃_{Y} → Y is obtained by taking the closed subscheme of X ×_{Z} Y defined by dividing out the O_{Z} torsion. In other words, if η is the generic point of Y, then X̃_{Y} is the closure of X ×_{Z} η in X ×_{Z} Y.

See [dJ], 2.18.

To compare varieties with projective varieties we have:

3.10. Chow’s Lemma. Given a variety X, there is a modification Y → X such that Y is quasi-projective.

See [M.RedBook] , I.10, p. 85, or [HAG], Exc. II.4.10 p. 107.

One way projectivity comes in handy is the following result, which uses the theory of Hilbert schemes:

3.11. Flattening Lemma. Given a variety Z (more generally, an integral scheme of finite presentation) and a projective morphism X → Z, there is a modification Y → Z such that the strict transform f′ : X̃_{Y} → Y is flat.

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

3.13. Definition. 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.

The entire workshop revolves around finding resolutions of singularities. To put things in prespective, we have:

3.14. Theorem (Hironaka). Suppose X is a variety over a fied k of characteristic 0. There is a sequence of blowings up X_{n} → ⋯ → X_{0} = X, each along a regular variety lying over SingX, such that X_{n} is regular. Thus X_{n} → X is a resolution of singularities in the strong sense.

See [H]. This type of results will be discussed in a parallel course in the workshop, taught by O. Villamayor.

We define a type of nice subvarieties:

3.15. Definition. 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 [dJ], 7.1.

4. Results

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

4.1. Theorem. 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.

4.2. Corollary. 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.

This result is completely new in positive characteristic. Notice that in characteristic 0, the result is an easy corollary of Hironaka’s theorem. Nevertheless, the mothods bring about new and important results in characteristic 0 as well.

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

4.3. Theorem (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 [dJ], 4.1.

De Jong’s theorem has a few important variants. First,

4.4. Theorem of semistable reduction up to alteration (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, the special fiber Y_{k_{1}} is a reduced, strict normal crossings divisor.

See [dJ], 6.5.

4.5. Theorem of equivariant resolution up to alteration (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 [dJ], 7.3.

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

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

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

4.7. Weak Resolution Theorem. Let X be a variety over an algebraically closed field of characteristic 0. Then there is a projective resolution of singularities (weak sense) Y → X.

4.8. Remark. This is a rather weak version of Hironaka’s theorem. The point is, that new proofs, by Abramovich and de Jong [ℵ-dJ], and by Bogomolov and Pantev [B-P], were given based on de Jong’s ideas. The proof by Bogomolov and Pantev is extremely simple, drawing only on toric geometry. The proof by Abramovich and de Jong lends itself to various generalizations in the flavor of de Jong’s semistable reduction theorem.

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

The next few sections discuss a few of the main toold involved in the proofs of the theorems.

5. Moduli of curves

The reader is referred to exercise set I for illuminations of the following definitions:

5.1. Definition. An S scheme C → S is called a nodal curve over S if it is proper and flat, such that all geometric fibers are connected curves with at most ordinary double points (locally xy = 0). (If S is allowed to be non-noetherian, we may also require C → S to be of finite presentation).

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

5.3. Definition (Deligne and Mumford). A nodal curve f : C → S, together with sections s_{i} : S → C, i = 1, …, n with image schemes S_{i} = s_{i}(S), is called a stable n-pointed curve of genus g if

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

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

3. All the geometric fibers have arithmetic genus g.

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

The definition is made so that a stable pointed curve has a finite automorphism group (relative over S). These definitions can be given in more down to earth terms - this will be done in the exercise sessions.

5.4. The functor of points and representability. We use the notion of reperesentable functors (as introduced by Grothendieck) in order to come to the definition of a fine moduli scheme with a universal family:

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 [M.RedBook], II§6.

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

F(−) ≅ Mor_{S}(−, M).

5.5. Moduli spaces - what are they? Suppose such a contravariant functor F is a moduli functor, i.e., it assigns to a scheme S the set of isomorphism classes of certain objects over S; e.g.

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

Assume F is representable by a scheme M. Then we 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)

is called a universal family.

It is a fact of life that several natural moduli functors in algebraic geometry are not representable. For this reason Mumford introduced the notion of a coarse moduli scheme, see [M.GIT], 5.2, requiring that the scheme gives geometrically what you want, and does it in the best possible way:

5.6. Definition. A scheme M and a functorial morphism

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 uniqe χ : N → M factoring ψ.

It can be seen that for any g ≥ 2 the functor M_{g} “of curves of genus g” is not representable, but it does admit a coarse moduli scheme. The same happens in the case of elliptic curves, where g = 1 with one marked point. We shall study some of these examples in the exercises.

5.7. Remark. Even if over (some open part of) a coarse moduli scheme there exsits a family giving the morphism φ : F → Mor_{S}(−, M) required in the definition of a cuarse moduli scheme, but the functor is not representable, we shall not call this family a universal family. Examples will be discussed.

5.8. Theorem/Notation. The functor M_{g} of smooth curves of genus g admits a coarse moduli scheme. We will denote it by M_{g} → Specℤ. The functor of smooth stable n-pointed curves of genus g is coarsely representable; notation: M_{g,n}. If n + 2g − 2 > 0, there is a coarse moduli scheme \overline{M_{g,n}}, projective over Spec Z, for stable n-pointed curves of genus g. The scheme M_{g,n} is open and dense in \overline{M_{g,n}}.

See [M.GIT], [D-M], [Knu].

Note that we write M_{g,0} = M_{g}.

The fact that \overline{M_{g,n}} → Specℤ is proper follows from what we call the “the weak valuative criterion”, which is based on the following extension result:

5.9. Theorem. Let S be the spectrum of a discrete valuation ring, η ∈ S the generic point. Let C_{η} → {η} be a smooth curve of genus g > 1. There exists a finite extension of discrete valuation rings S ↪ S_{1}, with generic point η_{1}, and an extension

\begin{matrix}C_{η_{1}} & ↪ & C \\ ↓ & & ↓ \\ \{η_{1}\} & ↪ & S_{1} \end{matrix}

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

5.10. Moduli schemes for curves with a level structure. We see that M_{g} → Specℤ is (only !) a coarse moduli scheme, which sometimes causes difficulties. Also M_{g,n} is coarse for small n, and \overline{M_{g,n}} is most often only a coarse moduli space. This may seem to be a serious drawback. To overcome that, Mumford has provided us with the following:

5.11. Definition. Let k be a field, and let m be a positive integer, not divisible by the characterisitc of k. A level-m-structure on a regular curve C over k of genus g consists of an isomorphism

α : Jac(C)[m] → (μ_{m})^{g} × (ℤ/m)^{g}

which is symplectic with respect to the Weil pairing; here Jac(C) is the Jacobian of C, and Jac(C)[m] denotes the m- torsion points. In case all m-th root of unity are in k we can write α as a symplectic ℤ/m-base {x_{1}, ⋯, x_{2g}} for Jac(C)[m].

5.12. Theorem. Suppose m ∈ ℤ_{≥3}. There is a fine moduli scheme

M_{g}^{(m)} → Spec Z[1/m]

for smooth curves over schemes on which m is invertible. The scheme

M_{g,n}^{(m)} → Spec Z[1/m]

is a fine moduli scheme for smooth n-pointed curves over schemes on which m is invertible.

5.13. Definition. Define \overline{M_{g}^{(m)}} → Z[1/m] to be the normalization of \overline{M_{g,n}} in the function field of M_{g}^{(m)}. Let \overline{M_{g,n}^{(m)}} be the normalization of \overline{M_{g,n}} in the function field of M_{g,n}^{(m)}. Let

π^{(m)} : \overline{M_{g,n}^{(m)}} → \overline{M_{g,n}} × Spec Z[1/m] be the natural finite morphism.

5.14. Remark. Note the somewhat strange approach: indeed we define schemes like \overline{M_{g}} by first defining a functor, and then showing it admits a coarse moduli scheme. However the schemes \overline{M_{g,n}^{(m)}} are defined by the somewhat mysterious process of first defining moduli schemes of stable curves via a coarsely representable functor, and then normalizing these in the function fields of moduli schemes of smooth curves with level structure. There is a good reason for this. We do not know an a priori obvious moduli-functor which defines the schemes like \overline{M_{g,n}^{(m)}}, this is semetimes expressed by saying: “we do not know a good moduli interpretation” of the points of \overline{M_{g,n}^{(m)}} . This will be discussed in the course.

5.15. Theorem. Suppose m ≥ 3. There exists a family of stable n-pointed curves

π^{(m)} : C^{(m)} → \overline{M_{g,n}^{(m)}}, with sections s_{i} : \overline{M_{g,n}^{(m)}} → C^{(m)},

such that π^{(m)} coincides with the moduli map of this family.

From this, de Jong deduces the following theorem, which in a sense resolves the singularities of the fibers of a family of curves:

5.16. Theorem. Let f : X → B be a projective morphism of finite type of integral schemes, Z ⊂ X a proper closed subset. Assume the generic fiber of this morphism is a smooth absolutely irreducible curve. Then there exists a diagram

X_{1} →^{f} X
π_{1} ↓  ↓
Σ ⊂ B_{1} → B

and sections s_{i} : B_{1} → X_{1}, satisfying:

1. B_{1} → B is an alteration;

2. X_{1} → X̃_{B_{1}} is a modification;

3. X_{1} → B_{1} is flat, projective and nodal;

4. s_{i} are disjoint sections, not meeting Sing(π_{1}); 5. Σ ⊂ B_{1} is a proper closed subset.

6. f^{−1}Z = ∪s_{i}(B_{1}) ∪ π_{1}^{−1}Σ.

6. Toroidal geometry

6.1. Definition. 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.

6.2. Definition. 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.

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 and D 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.

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.

A morphism between toric varieties is called a toric morphism, if it is surjective and torusequivariant. 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.

6.3. The polyhedral complex. Recall that there is a correspondence between

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

Here N may be canonically identified with the group of 1-parameter subgroups on T.

Since every toroidal embedding U ⊂ X looks locally like a toric variety, we can associate a cone σ_{x} to (the neighborhood of) a point x ∈ X. However, since we don’t have a fixed torus around, there is a slightly different definition of a linear embedding of this cone - it is embedded in N_{σ}, the dual of the group of Cartier divisors on a neighborhood of x, supported on X^{\} U. When U ⊂ X is actually toric, then N_{σ} ⊂ N is the subgroup spanned by σ.

Recall also, that given a general toric variety T ⊂ X, we can glue together the cones σ attached to toric affine open sets V_{σ} ⊂ X and form a fan Δ_{X} ⊂ N. This is in fact a one-to-one correspondence: we can build T ⊂ X back from the data Δ_{X} ⊂ N.

On the toroidal side we can do the same - we can glue together the cones of open sets in X. However, since they do not all lie canonically in one space N, what we get is not a fan - it is a rational conical polyhedral complex, also denoted Δ_{X}. Here we have no chance of getting a one-to-one correspondence.

Recall that there is a one-to-one correspondence between (proper) toric modifications T ⊂ X′ → X and (complete) subdivisions Δ′_{X} → Δ_{X}. The proof of this goes through word-for-word in the toroidal case - so proper “allowable” toroidal modifications correspond with subdivisions. Also, it is not difficult to describe when such a modification is projective.

Finally, recall that an affine toric variety is nonsingular if and only if the corresponding cone is generated by a subset of an integral basis for N. The same works in the toroidal case, if we replace N by the appropriate group as mentioned above.

The following theorem is thus relatively easy to prove:

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

See [KKMS], theorem 11*.

Part 2. Exercises

0. Methods of proof: a guide to the exercises

One striking feature of the proofs of de Jong’s theorem and its derivatives is, that all the ingredients, with the exception of one beutiful subtle result (the Three Point Lemma), were known and understood nearly two decades before. The way they are put together is quite ingenious. The purpose of this section is to help you understand how they are put together, once you put the effort into solving the exercises.

0.1. First steps. The proof of de Jong’s theorem starts with a series of simple reduction steps. These are discussed in the exercises in section I. Thus one may assume that the variety X is projective and normal, and the subset Z is the support of a Cartier divisor. Moreover, one may replace Z by a larger divisor. Using a fairly standard projection argument one may assume that we have a morphism X → P which makes X into a generically smooth family of curves, satisfying some nice properties, in particular Z → P is finite and unramified. Taking a separable alteration of the base P, we may assume that Z is the union of sections s_{i} : P → X.

0.2. 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 5 for discussion. Many exercises are available in sections II, III and IV. Using this theory, it is easy to see that one can find a diagram as follows:

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

where X_{1} → P_{1} is a family of stable pointed curves, where the “points” arise from the sections coming from Z. But in order for the inductive argument to work one needs β : X_{1} −−→ X to be an alteration, not a rational map! Here comes the following result:

0.3. Three Point Lemma (de Jong). Suppose Z meets the smooth locus of every irreducible component of every fiber in at least three points. Then β : X_{1} → X extends to a morphism.

See exercises in section V.

Once this lemma is proven, one simply needs to use induction on the dimension in order to resolve the base P and make the degeneracy locus into a divisor of normal crossings. A simple set of blowups (see exercises in section VI) finish the proof of theorem 4.3.

0.4. The variant theorems. Proving the variant theorems 4.4 and 4.5 goes by relatively easy modifications of the proof of 4.3. These modifications are discussed in VII. Basically, some of the good properties of the projection X → P used in the proof of 4.3 do not necessarily hold, therefore a little more work needs to be done. An important ingredient in the proof of 4.5 is the uniqueness of stable reduction of pointed curves.

0.5. Weak resolution in characteristic 0. Both new proofs of the weak resolution theorem start with reduction steps as in de Jong’s theorem. Once the projection is produced, the methods diverge.

The proof by Bogomolov and Pantev, discussed in exercises in section IX relies on a variation of a lemma of Belyi. One arrives at a situation where X is a finite cover of a smooth variety P branched over a divisor of normal crossings. It follows that X carries a toroidal structure, and therefore can be resolved easily. In a way, this is a generalization of the old method of Jung (also known as “Abhyankar’s dream” for resolutin in positive characterisic).

The proof by Abramovich and de Jong follows de Jong’s theorem more closely. One uses the stable reduction method to cover X → P by a family of nodal curves X_{1} → P_{1}. Some easy steps allow to make this into a finite Galois cover. After modifying the base P one may assume that X_{1} carries a toroidal structure. A simple argument provides an equivariant modification of X_{1} such that X_{1}/G is toroidal, and therefore can be resolved.

Both methods of proof seem to come close to resolving singularities in positive characteristic up to a purely inseparable alteration, but both lack crucial steps to produce such a result!

I. Simple reduction steps for de Jong’s theorem

I.1. Suppose you want to prove de Jong’s theorem 4.3. First you want to perform some elementary reductions:

1. Show that you may replace X by a modification X′ → X while replacing Z by its inverse image in X′.

2. By blowing up, show that you may assume that Z is the support of a Cartier divisor.

3. If Z_{i} ⊂ X are the supports of a Cartier divisors and Z_{1} ⊂ Z_{2}, show that to prove the theorem for (X, Z_{1}) it suffices to prove it for (X, Z_{2}).

4. Show, using Chow’s Lemma, that we may assume X is projective.

5. Show, using the first exercise, that we may assume X is normal.

I.2. The next step is to produce a projection with some nice properties. Let X ⊂ ℙ^{N} be a projective variety of dimension n.

1. Assume dim X < N − 1. Show that, if p is a general point in ℙ^{N}, then the projection of X through p to ℙ^{N−1} is finite and birational onto its image.

2. Assume dim X = N − 1. Show that if p is a general point in ℙ^{N}, then the projection of X through p to ℙ^{N−1} is finite and generically étale onto ℙ^{N−1}.

3. In the situation Z ⊂ X in the theorem, show, using a projection through a general plane of dimension N − n, that there is a finite set S of regular closed points in X^{\} Z, such that if X̃ is the blow up of X at S, then there is a morphism f : X̃ → ℙ^{n−1}, satisfying the following conditions:

(a) f is equidimensional of relative dimension 1

(b) The smooth locus of f is dense in all fibers

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

(d) At least one fiber of f is smooth

(e) The morphism f has connected fibers.

I.3. For the sake of the Three Point Lemma, it will be useful to enlarge Z so it meets every fiber sufficiently. This is done as follows:

1. Let X be a projective variety, C ⊂ X an irreducible curve, L a very ample line bundle. Let ℙ = ℙ(H^{0}(X, L^{⊗n})) and let ϕ_{n} : X → ℙ be the associated morphism. Then the image of C does not lie in any linear projective subspace of dimension < n.

2. Now let X be as in the theorem, with a projection X → P as produced above. Let C ⊂ X be an irreducible component of a geometric fiber of X → P. Show that there is a divisor H ⊂ X such that

(a) f|_{H} : H → P is finite and generically étale, and

(b)

#sm(X/P) ∩ C ∩ H ≥ 3.

(Here we count the points without multiplicities.)

3. apply Noetherian induction to obtain:

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

(a) f|_{H} : H → P is finite and generically étale, and

(b) for any irreducible component C of a geometric fiber of f, we have

#sm(X/P) ∩ C ∩ H ≥ 3.

Again, we count the points without multiplicities.

II. Exercises on moduli of curves

II.1. Definition (Deligne - Mumford). Let k be an algebraically closed field, and let C be an algebraic curve defined over k. Suppose that C is complete and connected. Let g ∈ ℤ_{≥2}. We say that C is a stable curve of genus g if:

1. dim_{k}(H^{1}(C, O_{C})) = g (i.e. C has arithmetic genus g),

2. the singularities of C are normal crossings of two regular branches (i.e. a point on C is either regular, or the completion of its local ring is on two parameters with equation x·y = 0),

3. if there is a component of C which is regular and rational (i.e. isomorphic to ℙ^{1}), then it has at least three points in common with other components.

II.2. Definition. Let K be a field, let k be an algebraically closed field containing K. Let C be an algebraic curve defined over K. We say that C is a stable curve if C ⊗ k is stable in the sense just defined.

II.3. Definition. A nodal curve C → S such that all fibres are stable (as defined above) is called a stable curve.

II.4. Exercise. Assume all conditions in the definition up to (2). Show that then (3) is equivalent by requiring:

3’. Aut(C) is a finite group.

II.5. Exercise. Show these definitions coincides with the one previously given.

II.6. Exercise. Give examples of stable curves, e.g. describe all “types” of stable curves of genus 3.

II.7. Definition (Deligne and Mumford, Knudsen). Let k be an algebraically closed field, and let C be an algebraic curve defined over k. Suppose that C is complete and connected. Let P_{1}, ⋯, P_{n} ∈ C(k) be points on C. Let dim_{k}(H^{1}(C, O_{C})) = g (i.e. C has arithmetic genus g). Suppose:

2g − 2 + n > 0.

We write S = Sing(C) for the set of singularities. We say that (C, P_{1}, ⋯, P_{n}) is a stable n-pointed curve of genus g if:

1. every singular point of C is a normal crossing of two regular branches,

2. for 1 ≤ i < j ≤ n we have P_{i} ≠ P_{j},

3. for every 1 ≤ i ≤ n we have P_{i} ∉ S,

4. if there is a component of C which is regular and rational, then then number of points either in S or among {P_{1}, ⋯, P_{n}} on that component is at least three.

II.8. Exercise. Assume all conditions in the definition up to (3). Show that then (4) is equivalent by requiring:

4’. Aut((C, P_{1}, ⋯, P_{n})) is a finite group.

II.9. Exercise. 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 such a generic curve has no nontrivial automorphisms.

3. (variant:) Show that such a general curve has no nontrivial automorphisms.

II.10. Definition. Let K be a field, let k be an algebraically closed field containing K. Let C be an algebraic curve defined over K, and let P_{1}, ⋯, P_{n} ∈ C(K). We say that (C, P_{1}, ⋯, P_{n}) is a stable n-pointed curve if (C, P_{1}, ⋯, P_{n}) ⊗ k is stable n-pointed in the sense just defined.

II.11. Exercise. 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).

II.12. Exercise. 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.

III. The j-line

In this exercise we analyse “moduli of elliptic curves” The motivation is: in history people looked for a long time for the right concept, it seems that Grothendieck with his notion of “representable functor” pinnned down the precise concept, but people soon found out that it did not give what you want in all cases^{1}.

Strictly speaking this set of exercises will not be used, but it can be used as a blueprint for al cases of moduli space of curves (without markings, without level structures).

Advice: if you know enough about ℓ-adic representations we suggest you substitute the exercise sets III, IV by more advanced problems and methods! For example see how to do the exercises mentioned, and also provide exercises and solutions for curves of arbitrary genus.

Let us do an easy case. 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}).

Note: for any elliptic curve, there is a general definition of the j-invariant (see [Silv]).

Suppose that E_{t} is given by the equation

Y^{2} = X^{3} + aX + a, with a = \frac{27·t}{4(1728 − t)}.

III.1. Exercise. For t ≠ 0, ≠ 1728 compute j(E_{t}). Conclude: there exists a familiy of elliptic curves

E → Spec(K[j, \frac{1}{j}, \frac{1}{j − 1728}]) ⊂ 𝔸_{K}^{1}

where each fibre E_{t} has the j-value equal to t.

However this family we do not call a “universal family”, it does not represent the functor of elliptic curves (see below).

III.2. ^{∗} Exercise. Show that any family

E → Spec(K[j, \frac{1}{j}, \frac{1}{j − 1728}]) ⊂ 𝔸_{K}^{1}

having the property j(E_{t}) = t cannot be extended to a family over the point t = 0 or t = 1728.

If you want, take K = ℂ, and use analytic methods, or do smething else... In the course we shall discuss methods which prove such statements “in general”.

III.3. ^{∗} Exercise. Let K = ℂ. How many families (up to isomorphisms) have the properties mentioned in (2) ?

Define a moduli functor for elliptic curves which do not have j = 0 or j = 1728, show it admits a coarse moduli space, show it does not admit a fine moduli space.

1See the introductory lines in the paper P. Samuel - invariants arithmétiques des courbes de de genre 2 [d’après Jun Ichi Igusa]. Sém. Bourbaki 14 (1961/62) N. 228: “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.”

III.4. Exercise. Let g ∈ ℤ_{≥3}. Show there exists a non-empty Zariski-open set U ⊂ M_{g} and a smooth curve C → U which gives the moduli map, i.e. for every geometric point u in U, the fibre C_{u} defines the moduli point [C_{u}] = u ∈ M_{g}. Show that once U is choosen, this family is unique.

IV. Stable reduction for elliptic curves.

In this exercise we illustrate the concept of stable reduction by studying the case of elliptic curves (i.e. curves of genus one with one marked point). 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.

IV.1. Notation. 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. 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).

IV.2. Exercise. 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.

We say that E has bad reduction at v if it does not have good reduction at v.

IV.3. Exercise. 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.

IV.4. Definition. 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.

IV.5. Exercise. Suppose R = k[S], with char(k) ≠ 2, ≠ 3, and let E overt 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.

IV.6. The case of Weierstrass equations. Suppose that E is given over K by a Weierstrass equation with coefficients in R (see [Silv]). Such an equation defines a plane curve E ⊂ 𝔸_{B}^{1} 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. If this 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. We say ”potentially stable” if we can find a model which has stable reduction over a finite extension.

IV.7. Exercise. 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 (and conclude that this curve has potentially bad reduction).

IV.8. Exercise. Let R be a DVR, with residue characteristic ≠ 2. Let E be an elliptic curve over K. Show this curve has potentially stable reduction !

IV.9. ^{∗} Exercise. Let R_{v} be a DVR, with residue characteristic ≠ 3. Suppose E is an elliptic curve over K given by a Weierstrass equation E = Z(F) ⊂ ℙ_{K}^{2} such that all flex points of E have coordinates in L ⊃ K, with w a discrete valuation of L over v. Show that E ⊗_{K} L has stable reduction at w.

IV.10. Conclusion. Every elliptic curve over a field K with a discrete valuation has potentially stable reduction at that valuation.

In the course we show this to be true for regular curves of arbitrary genus.

IV.11. ^{∗} Exercise. 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}. - Formulate (and prove?) the same result for curves of arbitrary genus. - If E is an elliptic curve over a field K, and End_{K}(E) ≠ ℤ, then E has potentially good reductions at every place of K.

V. Exercises on the Three Point Lemma

V.1. Performing stable reduction. Let X → P be a family of curves satisfying the conditions in I.2. Suppose Z ⊂ X satisfies the conditions in I.3. We first perform stable reduction to this situation

1. Show that, by replacing P by a separable alteration and X by the strict transform, we may assume that there are sections s_{i} : P → X such that Z is the union of their images.

2. Use theorem 5.15 to show that there exists an alteration P_{1} → P, a family of pointed stable curves C → P_{1}, τ_{i} : P_{1} → C and an open set U_{1} ⊂ P_{1} with an isomorphism C|_{U_{1}} → X ×_{P} U_{1} carrying τ_{i} to σ_{i}.

We may as well replace P by P_{1} and X by its strict transform. We thus have a birational map C −−→ X carrying τ_{i} to σ_{i}. We want it to extend to a morphism, at least after a modification of P!

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

1. Use the flattening lemma to show that 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.

2. Let p ∈ P be a geometric point, X_{p}, T_{p}, C_{p} the fibers over p. Show that there exists a finite set W ⊂ X_{p} such T_{p} → X_{p} is finite away from W.

3. If x ∈ X_{p} ^{\} W, conclude that there is an open set V ⊂ X such that pr_{2}^{−1}V → V is finite and birational.

4. If x ∈ sm(X_{p})^{\} W show that we may choose V to be normal; thus pr_{2}^{−1}V → V has to be an isomorphism in this case.

5. Show that if X′ ⊂ X_{p} is an irreducible component, then there is a unique irreducible component T′ of T_{p} mapping finitely and birationally onto X′ via pr_{2} : T → X.

6. By repeating the argument for pr_{1} : T → C, sho that if C′′ ⊂ C_{p} is an irreducible component, then there is a unique irreducible component T′′ of T_{p} mapping finitely and birationally onto C′′ via pr_{1} : T → C.

7. Let x ∈ sm(X_{p}) be a closed point. By considering the Stein factorization T → X̃ → X, show that the fiber pr_{2}^{−1}(x) is connected.

V.3. Using the three point assumption. Let X′ ⊂ X_{p} be an irreducible component, and T′ ⊂ T_{p} the unique component mapping finitely onto it, as in exercise 5 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 σ_{i} : P → X, i = 1, …, 3 be three of the given sections such that σ_{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}.

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

2. Assume c ∉ {c_{i}, i = 1, …, 3}. Show that each of T_{i}, i = 1, …, 3 contains an irreducible component T′_{i} whose image in C is again a curve passing through c. Show that these image components are distinct. This contradicts the fact that C_{p} is nodal.

3. Assume, without loss of generality, c = c_{1}. Repeat the argument of the previous exercise to conclude that there are two distinct components of C_{p} passing through c, contradicting the fact that C_{p} has a marked point at c = c_{1}.

4. Conclude that pr_{1} : T → C is finite and birational.

5. Show, using Serre’s criterion, that C is normal (condition S_{2} follows since C → P has reduced 1-dimensional fibers and P is normal).

6. Conclude that T → C is an isomorphism, hence β extends as a morphism.

VI. Exercises on elementary resolution of nice nodal families

An important idea in de Jong’s work, is that once one has a variety X mapping onto a variety S as a nice family of curves over a nice base, it is easy to resolve its singularities. The purpose of these exercises is to go through the resolution in such a case.

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 now 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.

VII. Modifications of the proof for the variant theorems

VII.1. First we consider some possible variants of the elementary reduction steps, and their consequences. Thus, consider the situation X → P we have obtained in the elementary reduction steps, see exercises in section I. We will try to see what happens if we drop some of the nice properties it had.

1. Using an alteration, show that the condition that Z → P be finite is unnecessary.

2. Show that, if one is willing to accept inseparable alterations in the theorem, the condition that Z → P be generically étale is unnecessary.

3. * By reviewing the arguments, show that the condition that every fiber of X → P be generically smooth is unnecessary. (Here a modification of the three point lemma may be necessary!)

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

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.

VII.2. Let us address the Theorem on semistable reduction up to alteration 4.4. Suppose S = Spec R where R is a discrete valuation ring, and X → S is a morphism as in the theorem, and Z a proper closed subset.

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

2. Show that we may assume that the generic fiber is a normal variety, and that X is a normal scheme.

3. ∗ Show that we may assume that the special fiber X_{s} is generically reduced.

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

5. Use the semistable reduction argument, with the Three Point Lemma, to replace X by a nice family of curves X → P → S.

6. Use induction on the dimension to conclude the proof of the theorem.

7. 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!)

VII.3. 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 4.5.

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

2. Show that we may assume Z is a divisor.

3. 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.

4. 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.

5. Consider the case X = 𝔸_{k}^{2} where chark = 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!

6. 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.

7. ∗ 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.

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

VIII. 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. Now 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 polyhedrral complex. (Compare with the fan of ℙ^{2}!)

8. Now 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!

IX. Weak Resolution according to Bogomolov and Pantev

Granted toroidal resolution, the proof of weak resolution 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 [B-P], which came up in discussions with T. Pantev. These and additional simplifications were discovered independently by K. Paranjape, and we have used his beautiful exposition in some of the following exercises.

IX.1. 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 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 mapping finitely to ℙ^{n−1}.

We now replace Z by Z ∪ f^{−1}B.

IX.2. The next steps are aimed at replacing P by another ℙ^{1}-bundle over ℙ^{n−1}, such that the branch locus of X → P 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.

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} (1).

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 another 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.

IX.3. 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 [B-P], and in Paranjape’s exposition) that this can be 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 now 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 exercise set VI, to replace Q by a nonsingular variety, such that the inverse image of D is a strict divisor of normal crossings.

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.

X. The Torific Ideal

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. 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.

X.1. 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. Defing I_{G} = ⟨M_{u}⟩, the ideal generated by M_{u}.

X.2. Exercise (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′}.

X.3. Exercise (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.

X.4. Exercise. Show that I_{G} is generated by u and a finite number of monomials t_{1}, …, t_{m} in M_{u} ∩ R.

X.5. Exercise. 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.

X.6. Exercise. 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}.

X.7. Exercise. 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.

References

[ℵ-dJ] D. Abramovich and A. J. de Jong, Smoothness, semistability and toroidal geometry, J. Alg. Geom.,

            to appear. alg-geom/9603018

[ℵ-W] D. Abramovich and J. Wang, Equivariant resolution of singularities in characteristic 0, Math.

Res. Letters, toappear. alg-geom/9609013

[A-W] M. Artin and G. Winters, Degenerate fibres and reduction of curves. Topology 10 (1971), 373-383.

[BerBour] P. Berthelot, Altérations de variétés algébriques [d’après A. J. de Jong]. Sém. Bourbaki 1995-96, Exp. 815.

[B-P] F. Bogomolov and T. Pantev, Weak Hironaka theorem, Math. Res. Lett. 3 (1996), no. 3, 299–307.

[Bry] J-L. Brylinski, Propriétés de ramification à l’infini du groupe modulaire de Teichmüller. Ann. Sci. Ecole Norm. Sup. (4) 12 (1979), 295-333.

[D] P. Deligne, Le lemme de Gabber. In: Sém. sur les pinceaux arithmétiques: la conjecture de Mordell (Ed. L.. Szpiro). Astérisque 127 (1985), 131-150.

[D-M] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus. Publ. IHES, No. 36 (1969) (Volume dédié au Professor Oscar Zariski à l’occasion de son 70^{e} anniversaire), pp. 75 - 109.

[E-H] D. Eisenbud and J. Harris, Schemes: the language of modern algebraic geometry. Wadsworth &

Brooks/Cole, 1992.

[F] W. Fulton, Introduction to toric varieties, Annals of Math. Studies 131. [vG-O] B. van Geemen and F. Oort, A compactification of a fine moduli spaces of curves. University of

Utrecht, Preprint 301, 1983.

[H-M] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves. Invent.

Math. 67 (1982), 43-70.

[Harris] J. Harris, Algebraic Geometry - a first course. Grad. texts in Math. 133, Springer - Verlag, 1992.

[HAG] R. Hartshorne, Algebraic geometry. Grad. texts Math. 52, Springer - verlag, 1977.

[H] H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero: I, II, Ann. of Math. (2) 79 (1964), 109-326.

[KKMS] G. Kempf, F. Knudsen, D. Mumford and B. Saint-Donat, Toroidal Embeddings I, Springer, LNM 339, 1973.

[Knu] F. F. Knudsen, The projectivity of the moduli space of stable curves, II: the stacks M_{g,n}. III: The line bundles on M_{g,n} and a proof of the projectivity of \overline{M_{g,n}} in characteristic zero. Math. Scand. 52 (1983). 161-199, 200-221.

[dJ] A. J. de Jong, Smoothness, semistability, and alterations, Publications Mathématiques I.H.E.S. 83, 1996, pp. 51-93.

[dJ2] A. J. de Jong, Families of curves and alterations, Annales de l‘Institute Fourier, to appear. [dJ-O] A. J. de Jong and F. Oort, On extending families of curves, J. Alg. Geom. 6 no. 3 (1997), 545-562.

[Mos] M. Mostafa, Die Singularitäten de Modulmannigfaltigkeiten \overline{M}_{g}(n) der stabilen Kurven vom Geslecht g ≥ 2 mit n-Teilungspunkstruktur. Journ reine angew. Math. (Crelle), 343 (1983), 81–98.

[M.C+J] D. Mumford, Lectures on curves and their jacobians. University of Michgan Press. [M.GIT] D. Mumford, J. Fogarty and F. Kirwan Geometric invariant theory. Erben. Math. N.Folge, Bd.

34, Springer - Verlag, 1994.

[M.RedBook] D. Mumford, The red book of varieties and schemes. Lect. Notes Math. 1358, Springer - Verlag

1988.

[O] F. Oort, Good and stable reduction of abelian varieties. Manuscr. Math. 11 (1974), 171 - 197.

[Po1] H. Popp, Moduli theory and classification theory of algebraic varieties. Lect. Notes Math. 620, Springer - Verlag 1965.

[Po2] H. Popp, On the moduli of algebraic varieites III, Fine moduli spaces. Compos. Math. 31 (1975), 237 - 258.

[Po3] H. Popp, The singularities of moduli schemes of curves. Journ. Number theory 1 (199), 90 - 107.

[Ray] M. Raynaud, Compactification du module des courbes. Sém. Bourbaki 23, (1970/71), Exp. 385. [S-T] J-P. Serre & J. Tate, Good reduction of abelian varieties. Ann. Math. 88 (1968), 492-517 [Serre

ŒII, 79].

[SGA] A. Grothendieck (with M. Raunaud and D. S. Rim), Groupes de monodromie en géométrie algébrique (SGA 7 I). Séminaire de géométrie algébrique du Bois-Marie. LNM 288, Springer - Verlag 1972.

[Silv] J. Silverman, The arithmetic of elliptic curves. Grad. texts in Math. 106, Springer - Verlag, 1986.