STABLE POINTED CURVES, ALTERATIONS AND APPLICATIONS.

FRANS OORT

Working week on resolution of singularities September 7 - 14, 1997, Obergurgl, Tirol, Austria. Survey of results discussed in my talk.

Introduction

Here a short survey of the results we shall discuss in my talks. The theorems mentioned in this note are the main results I am going to state in my talks. Unfortunately I will not have enough time to discuss all proofs in full detail. In every lecture I shall state at least one clear statement. Some details of its proof will be discussed. Some other parts of the proof we will omit (however enough motivation and references are available).

Terminology and notations as explained in

Dan Abramovich & Frans Oort -

- Alterations and resolution of singularities, (informal notes) cited as [DA.FO].

will be used freely.

0.1. Motivation: One of the basic ideas of removing singularities via alterations as started by A. J. de Jong is: fibre a variety Y as a family of curves: Y → X, and change X and Y (via alterations) in such a way that all fibres have only “easy” singularities (X′ is nodal over Y′).

0.2. Basic Question: Here is the central theme of my talk:

• Given a scheme S and a dense open subset U ⊂ S, and

• and a family of curves C_{U} → U (think of a smooth curve over U, or a nodal curve for example),

• does there exist a curve C → S extending this ?? By “extending” we mean: the restriction of C → S to U is isomorphic with C_{U} → U. The question can be asked e.g. in case C → S is smooth, or nodal or something alike. [In the proof of (3.1) we need a stronger variant of this question, see [13], 4.18.]

In general the answer, even to the weakest form of this question, is negative! We shall discuss various situations and examples, and see what can be achieved (extending after alterations, extending in certain good situations), we will start the talk by giving various (counter)examples.

By the way, do not worry about “schemes” and so on. If you are accustomed to work with plain varieties over ℂ, then keep doing so, and you can still be perfectly happy with the contents of my talk. But for those of you who prefer to think in mixed characteristics, or over arbitrary base fields, I hope the extra data in my theorems will be clear.

Date: September 20, 1997.

Here is a situation which is very instructive and basic for what follows: Suppose moreover that the dimension of S equals one, and that S is nonsingular (and, if you wish, U is the complement of one closed point). In this case the extension problem has a positive answer, and even admits a “best” answer:

0.3. Minimal models, existence and uniqueness: Let K be a field, and C a complete, regular, geometrically irreducible algebraic curve defined over K; suppose the genus of C is at least 2. Let v be a discrete valuation on K, and R ⊂ K its ring. Using resolution of singularities and following the ideas of Zariski of studying minimal models of algebraic surface (1956) we arrive at minimal models: using ideas and methods of Abhyankar (1963) we can resolve surface singularities, by Shafarevich (1966) we have the notion of the minimal model of C over S := Spec(R) (see Lichtenbaum, [17], Th. 4.4; also see [6], page 87): we arrive at a curve C → Spec(R) which is a regular 2-dimensional scheme, and which is relatively minimal. Remark: here we use a special case of “resolution of singularities”, namely in the case of “arithmetic surfaces”.

1. Stable curves.

Given a family of curves C′ → U over an open set U ⊂ S we like to extend this to some reasonable family C → S. (It is a fact of life that this is sometimes not possible, we shall discuss that later.) Even if such an extension is possible, the “best” extension in general is not a nodal family. Moreover, in general the “best extension” does not commute with base change. This causes the phenomenon that “taking the limit” of a regular curve is not unique in a geometric sense. If we would try to construct a moduli space of all curves (of a given genus, regular or singular), we would arrive at non-Hausdorff moduli spaces! However there is a good notion which singular curves should be admitted, enough to assure that these appear after an extension of the base field, geometrically: after a (ramified) extension. In this way we arrive at compactifications of moduli spaces which are “compact”.

Historically, the influential paper [6] 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 [20] we see that in 1964 Mumford was trying to find the appropriate notions assuring good compactifications.

For the definition of a “stable curve” see [DA.FO], 7.1, 5.3 and 7.7 (we shall use that the two definitions given there are equivalent). A “curve” will be an algebraic curve over a field, complete (proper over that field) and geometrically connected. A “family of curves” will be a flat, proper morphism C → S such that all geometric fibres are algebraic curves; sometimes we say “C is a curve over S”. We recall the stable reduction theorem for curves, see [DA.FO], 5.9:

1.1. 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_{1} \\ ↓ & & ↓ \\ \{η_{1}\} & ↪ & S_{1} \end{matrix}

such that C_{1} → S_{1} is a family of stable curves. See [6], page 92, Coroll. 2.7. See [1], Coroll. 2.10.

In slightly different terminology: (1.1’): Consider a field K, with a discrete valuation v, and an algebraic curve C defined over K; suppose C is regular (and complete and geometrically irreducible) of genus g > 1. Then there exists a finite extension K ⊂ K_{1}, a valuation v_{1} with ring R_{1} ⊂ K_{1} extending v and an extension of C ⊗_{K} K_{1} to a stable curve over R_{1}.

Note that the curve C defined over K and the curve C_{1} defined over K_{1} are indicated by different symbols (in classical literature you will often find them given by the same symbol, confusion is possible!).

1.2. Remark: Suppose C = C_{η} → {η} extends to C → S, a minimal model of C, and C_{1} → S_{1} is the minimal model of C_{1} := C ⊗ K_{1}; in general

C ×_{S} S_{1} → S_{1} is not isomorphic with C_{1} → S_{1}.

In fact, if C → S is not stable, and C_{1} → S_{1} is stable (as in the situation of the theorem) this is clear. However if K_{1}/K is unramified at v, then C ×_{S} S_{1} → S_{1} and C_{1} → S_{1} are isomorphic.

There are (at least) two basically different proofs of the stable reduction theorem. There is a proof by M. Artin and G. Winters, see [1], using matrices of intersection numbers of components in special fibres, also see [27]. In the course we shall follow the proof by Grothendieck (based on [SGA]), see [6], §2.

1.3. Here is a (short) description of the proof which is based on ideas by Grothendieck. Let C be a curve smooth over a field K, and J := Pic_{C}^{0} its Picard variety (also called its Jacobi variety, “equal” to its Albanese variety if C(K) ≠ ∅). Here is the first of the three basic ideas:

(I) The curve C has stable reduction at a given discrete valuation v of K iff J has stable reduction at v (concepts will be defined in the course), see [6], Th. 2.4.

Choose a prime number ℓ different from the residue characteristic of v, and consider the Tate-ℓ-group T_{ℓ}(X) of an abelian variety X over K (later we choose J = X).

(II) We know that X has stable reduction if and only if the local monodromy

ρ_{X,ℓ} = ρ : I_{v} → Aut(T_{ℓ}(X))

is unipotent, see [SGA], page 21, Th. 6.1; see [SGA], IX.3, especially Prop. 3.5 on page 350, and Coroll. 3.8 on page 353, for a proof using [1]; also see [22], pp. 22-26, see [16], Prop. (1.3). We say that an abelian variety has stable reduction at v, if its Néron minimal model X at v has a special fibre whose connected component X_{v}^{0} is an extension of an abelian variety by an algebraic torus.

As Grothendieck showed:

(III) eigenvalues of algebraic monodromy are roots of unity

(see [30], appendix, page 515; see [22], (1.3); see [23], 7.4).

Conclusion: there exists a finite extension K ⊂ L, and a discrete valuation w of L over v such that:

the eigenvalues of ρ_{X,ℓ} are roots of unity ⟹

the eigenvalues of ρ_{X_{L},ℓ} are equal to 1 ⟹

X_{L} has stable reduction at w ⟹

(take X = J := Pic_{C}^{0})

C_{L} has stable reduction at w.

1.4. Warning: We see that

J has stable reduction at v ⟹ C has stable reduction at v,

however

J has good reduction at v ≠⇒ C has good reduction at v.

1.5. Remark, not used in the course: In order to test properties of good reduction of an algebraic curve over a discrete valuation ring using properties of monodromy, one can consider monodromy actions on the fundamental group of an algebraic curve, see [2]. We have:

• An abelian variety has stable reduction iff the monodromy representation (on the Tate-ℓ-group, or on homology) is unipotent, it has good reduction iff this monodromy representation is trivial.

• An algebraic curve has stable reduction iff the monodromy representation (on homology, or on the Tate-ℓ-group of its Jacobian) is unipotent,

• it has “compact type of reduction” (i.e. its Jacobian has good reduction, i.e. we can choose a reduction to a nodal curve which is a tree of nonsingular components) iff this representation is trivial.

• An algebraic curve has good reduction iff the monodromy representation by outer automorphisms on the fundamental group is trivial.

2. Moduli spaces.

2.1. Motivation. In order to be able to use stable curves, and compactified moduli space in order to extend families of curves we need existence of families of curves over a compactified moduli space in such a way that the moduli point of a fibre is that point in the moduli space. Warning: we use the terminology “universal family” only in case we have a representable functor; in what we need we shall often speak of a “tautological family”, or a ”tautological curve”. We recall theorems proved in [GIT], [6], and in [15]; see also [9].

2.2. 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).

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

M_{g,n}^{(m)} → Spec ℤ[1/m],

and a universal family

π^{(m)} : C^{(m)} → M_{g,n}^{(m)}, with sections s_{i} : M_{g,n}^{(m)} → C^{(m)}, 1 ≤ i ≤ n,

representing the functor of stable n-pointed smooth curves of genus g, with a level m-structure over schemes on which m is invertible.

2.4. 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)}, 1 ≤ i ≤ n,

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

2.5. Remark, exercise: Using the valuative criterion of properness, see [HAG], Th. II.4.7, and using the stable reduction theorem for curves, see (1.1), and by methods from [15], also see [12], Section 4, we conclude that

\overline{M_{g,n}^{(m)}} → Spec ℤ[\frac{1}{m}]

is a proper morphism for every m ∈ ℤ_{≥1}, and every n ∈ ℤ_{≥0}. Remark: one can even show (Mumford, Knudsen), that this morphism is projective.

2.6. Exercise: Let us consider a Zariski-dense open set U ⊂ M_{g,n}^{(m)}, and take a family of curves C_{U} → U (possibly with extra structures like level and marked points). We say that C_{U} → U is a tautological curve if for every geometric u ∈ U the moduli point u′ ∈ M_{g} (image of u) is the moduli point of the fibre C_{U} ×_{U} {u}. (Sometimes C_{U} → U with these properties is called a “universal family”, a terminology which I find confusing.) [In this exercise you can either work over ℤ, or over a field, as you like!]

a) Show that there exists a dense open set U ∈ M_{1,1} (here m = 1) which carries a tautological curve; show that this tautological curve is not unique (up to isomorphism over M_{1,1}).

b) Show that no dense open set U ∈ M_{2} (here n = 0, m = 1) carries a tautological curve (see [32], page 177, Theorem 3).

c) For every g ∈ ℤ_{≥2} show the existence of a dense open subset U ∈ M_{g} (here n = 0, m = 1) carrying a tautological curve. Show that this is unique. Define the moduli functor for all smooth curves of genus g, where g ∈ ℤ_{≥2} is fixed, considering only curves which have no non-trivial automorphisms (even after extending base fields); show that this functor is representable.

2.7. Remark, exercise: Let S be a Noetherian scheme, and U ⊂ S a dense open subset. Let C_{U} → U be a stable curve. There exists an alteration T → S, with U′ := U ×_{S} T and D_{U′} := C_{U} ×_{U} U′ and a stable curve D → T extending D_{U′} → U′. [This will be used in the proof of (3.1), but there we shall need a refinement of this exercise.]

3. Alterations can remove singularities.

We recall the central theorem by A. J. de Jong which will be proved in the course:

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

f^{−1}Z ↪ Y ↪^{j} \overline{Y}
↓ ↓f
Z ↪ X.

See [13], 4.1.

4. Intersection multiplicities (a conjecture by Serre).

Intersection theory has a long history (and certainly we are not going to say much about it here). One aspect: it is not so easy to have a good definition for intersection multiplicities.

4.1. Remark, exercise: let C ⊂ ℙ_{k}^{2} be a plane algebraic curves, P ∈ C a closed point at which C is regular, and D = Z(F) ⊂ ℙ^{2} a plane curve given by a homogeneuous 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.

4.2. 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.

4.3. Exercise (See Gröbner [10], 144.10/11, also see [29], [28], [3], see [HAG], I.7):

a) Let C ⊂ ℙ^{3} be the space curve with parametrization

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

(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).

4.4. 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 [29], also see [3], 6.1, see [28]), 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 mutiplicities etc.).

4.5. Theorem (Conjectured by Serre, proved by Gabber, see [3], 6.1): 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.

5. Extending families of curves.

Here is a basic question: given a scheme S, an open dense subset U ⊂ S, a family of curves C_{U} → U, can this be extended to a family C → S (having certain reasonable properties). An analogous question can be asked for families of abelian varieties, for other types of families. These seem difficult questions.

5.1. Lemma: Let S be a normal locally Noetherian scheme, U ⊂ S a dense open subset, and C_{U} → U a stable curve. Then there exists at most one stable curve C → S which extends C_{U}. See [14], 2.3 for a more general statement; for a statement concerning semiabelian schemes, see [7], page 350, Lemma 1, see [8], page 9, Prop. 2.7.

Using the theory of stable curves we can show an extension theorem for families of curves, to be found in [14]. To this end we define:

5.2. Definition: Let C be an n-pointed stable curve over an algebraically closed field k. The topological type of this pointed curve is a graph (the “dual graph”) where edges are labeled with pairs of non-negative integers, given by:

• Every irreducible component of C corresponds with a vertex,

• the genus of that component appears as first label at that vertex, the number of marked points on that component as second label.

• Every singularity of C is a node, and it corresponds with an edge of the graph; the edge has as endpoints the component(s) on which the singularity is situated (NB if the node is a singularity of one component, the edge starts and and ends in the same vertex).

There is a slight difference with the definition as on page 86 of [6], here every singularity gives an edge, in [6] only singularities situated on two different irreducible components define an edge. For example, an irreducible curve with one singular point (a node) defines a graph with one edge and one vertex, while in [6] it corresponds with a graph with one vertex, and no edges.

5.3. Definition: Let C_{U} → U be a family of stable n-pointed curves. We say that this family is locally of constant topological type if for every two geometric points u_{1}, u_{2} ∈ U the fibres have equal topological types (i.e. the dual graph with labelling is constant on geometric fibres).

5.4. Theorem (see [14], Th. 5.1): Let S be a regular scheme, D ⊂ S a normal crossings divisor, and U := S − D. Let C_{U} → U be a stable curve which is locally of constant topological type, such that C_{U} extends to a stable curve over the generic point of every component of D. Then C_{U} → U extends to a stable curve over S.

5.5. Remarks: It is easy to give a surface S, a closed point P ∈ S, such that S − {P } is regular (but S singular at P), and a smooth curve C^{0} → S − {P } which cannot be extended to any curve over S, i.e. the condition “S is regular” is needed. There are examples which show that without the condition “D is a normal crossing divisor” the conclusion of the theorem need not be true.

5.6. Key Lemma (see [14], 4.1): Let k be an algebraically closed field, let C → ℙ_{k}^{1}, with s_{i} : ℙ_{k}^{1} → C be a family of stable n-pointed curves which is locally of constant topological type when restricted to U := ℙ_{k}^{1} − {0, ∞}. Then this family is constant i.e. take the pointed stable curve (C_{1}, {s_{i}(1)} over 1 ∈ U, then

(C, {s_{i}}) ≅ (C_{1}, {s_{i}(1)}) ×_{Spec(k)} ℙ_{k}^{1}.

5.7. Remarks: There exist smooth pointed curves over U := ℙ_{k}^{1} − {0, ∞} which are not constant, i.e. without the condition “(C, {s_{i}}) is stable over ℙ_{k}^{1}” the conclusion of the lemma does not hold in general.

For families of stable curves over ℙ_{k}^{1} with at most two singular fibres: see the result by L. Szpiro, [33], Th. 4 on page 61. For extension problems in codimension 2, and for certain families over ℙ_{k}^{1}, see [18].

For a discussion and an example (by Raynaud-Ogus-Gabber) on the extension problem for families of abelian varieties, see [14], Section 6.

References

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

[2] M. Asada, M. Matsumoto & T. Oda, Local monodromy on the fundamental groups of algebraic curves along a degenerated stable curve. Journ. Pure Appl. Algebra 103 (1995), 235–283.

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

[4] C. Chevalley, Une démonstration sur les groupes algébriques. Journ. math. pures appl. (Crelle), 39 (1960), 307–317.

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

[6] 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.

[7] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. Invent. Math. 73 (1983), 349 – 366.

[8] G. Faltings & C.-L. Chai, Degeneration of abelian varieties. Ergebnisse Math. Bd. 22, Springer - Verlag, 1990.

[9] B. van Geemen and F. Oort, A compactification of a fine moduli spaces of curves. University of Utrecht, Preprint 301, 1983.

[10] W. Gröbner, Moderne algebraische Geometrie. Springer - Verlag, 1949.

[11] W. J. Haboush, Reductive groups are geometrically reductive. Ann. Math. 102 (1975), 67 – 83.

[12] J. Harris & D. Mumford, On the Kodaira dimension of the moduli space of curves. Invent. Math. 67 (1982), 23–86.

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

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

[14] A. J. de Jong and F. Oort, On extending families of curves, Journ. Alg. Geom., to appear.

[15] 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.

[16] H. W. Lenstra Jr. & F. Oort, Abelian varieties having purely additive reduction. Journ. Pure Appl. Algebra 36 (1985), 281–298.

[17] S. Lichtenbaum, Curves over discrete valuation rings. Amer. Journ. Math. 90 (1968), 380–405.

[18] L. Moret-Bailly, Un théorème de pureté pour les familles de courbes lisses. C. R. Acad. Sc. 300 (1985), 489–492.

[19] 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.

[20] D. Mumford, The boundary of moduli problems. In: Lect. Notes Summ. Inst. Algebraic Geometry, Woods Hole 1964; 8 pp.

[21] D. Mumford, Stability of projective varieties. Enseign. Math. 22 (1977), 39 – 110.

[RedBook] D. Mumford, The red book of varieties and schemes. Lect. Notes Math. 1358, Springer - Verlag 1988.

[GIT] D. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory. Ergebn. Math. N. Folge, Bd. 34, Springer - Verlag, 1994.

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

[23] F. Oort, The algebraic fundamental group. In: Geometric Galois actions (Ed. L. Schneps & P. Lochak), London Math. Soc. Lect. N. 242, Cambridge Univ. Press, 1977; 67 - 83.

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

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

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

[27] M. Raynaud, Compactification du module des courbes. Sém. Bourbaki 23 (1970/71), Exp. 385.

[28] P. C. Roberts, Intersection theory and the homological conjectures in commutative algebra. Proceed. ICM, Kyoto 1990. Math. Soc. Japan & Springer - Verlag, 1991; Vol. I, pp. 361 - 368.

[29] J-P.Serre, Algèbre locale - multiplicités. Lect. Notes Math. 11, Springer - Verlag 1961.

[30] J-P. Serre & J. Tate, Good reduction of abelian varieties. Ann. Math. 88 (1968), 492-517 [Serre ŒII, 79].

[31] C. S. Seshadri, quotient spaces modulo algebraic groups. Ann. Math. 95 1971, 511 – 556.

[32] G. Shimura, On the field of rationality for an abelian variety. Nagoya Math. Journ. 45, (1971), 167 – 178.

[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. Lect. Notes Math. 288, Springer - Verlag 1972.

[33] L. Szpiro, Séminaire sur les pinceaux de courbes de genre au moins deux. Astérisque 86, Soc. Math. France, 1981.

Frans Oort

Mathematisch Instituut Budapestlaan 6 NL - 3508 TA Utrecht The Netherlands email: oort@math.ruu.nl