THE AFFINE STRATIFICATION NUMBER AND THE MODULI SPACE OF CURVES

MIKE ROTH AND RAVI VAKIL

Abstract. We define the affine stratification number asn X of a scheme X. For X equidimensional, it is the minimal number k such that there is a stratification of X by locally closed affine subschemes of codimension at most k. We show that the affine stratification number is well-behaved, and bounds many aspects of the topological complexity of the scheme, such as vanishing of cohomology groups of quasicoherent, constructible, and ℓ-adic sheaves. We explain how to bound asn X in practice. We give a series of conjectures (the first by E. Looijenga) bounding the affine stratification number of various moduli spaces of pointed curves. For example, the philosophy of [GV, Theorem ⋆] yields: the moduli space of genus g, n-pointed complex curves of compact type (resp. with “rational tails”) should have the homotopy type of a finite complex of dimension at most 5g − 6 + 2n (resp. 4g − 5 + 2n). This investigation is based on work and questions of Looijenga.

One relevant example (Example 4.9) turns out to be a proper integral variety with no embeddings in a smooth algebraic space. This one-paragraph construction appears to be simpler and more elementary than the earlier examples, due to Horrocks [Ho] and Nori [N].

Contents

1. Introduction 1 2. Basic properties of affine stratifications 3 3. Reorganizing affine stratifications 6 4. Topological consequences of bounded affine stratification number 10 5. Applications to moduli spaces of curves 15 References 17

1. Introduction

The affine stratification number of a scheme X bounds the “topological complexity” of a scheme. For example, it bounds the cohomological dimension cd X of X, which is the largest integer n such that H^{n}(X, F) ≠ 0 for some quasicoherent sheaf F (Proposition 4.12). Similarly, the cohomology of any constructible or ℓ-adic sheaf vanishes in degree greater than asn X + dim X (Proposition 4.19). We expect that if the base field is ℂ, then X has the

Date: Friday, June 18, 2004. The first author is partially supported by NSERC. The second author is partially supported by NSF grant

DMS–0228011 and NSF CAREER grant DMS–0238532.

2000 Mathematics Subject Classification: Primary 14A15, Secondary 14H10.

homotopy type of a finite complex of dimension at most asn X + dim X (Conjecture 4.21), but have not completed a proof. (Unless otherwise stated, all schemes and stacks are assumed to be separated and of finite type over an arbitrary base field.)

A related, previously studied invariant is the affine covering number acn X, which is one less than the minimal number of affine open sets required to cover X. The affine stratification number is bounded by acn X, is better behaved (e.g. is bounded by dimension, cf. Example 4.8), and has the same topological consequences. We know of no interesting consequences of bounded acn that are not already consequences of the same bound on asn.

For equidimensional X, the definition is particularly simple.

1.1. Definition. The (equidimensional) affine stratification number of an equidimensional scheme X is the minimal number easn X such that there is a (finite) stratification of X by locally closed affine subschemes of codimension at most easn X.

This is the form most likely to be of interest. The appropriate generalization to arbitrary schemes is only slightly more complicated.

1.2. Definition. An affine stratification of a scheme X is a finite decomposition X = ⨆_{k∈ℤ^{≥0},i} Y_{k,i} into disjoint locally closed affine subschemes Y_{k,i}, where for each Y_{k,i},

(1)  \overline{Y}_{k,i} \ Y_{k,i} ⊆ ⋃_{k′>k, j} Y_{k′,j}.

The length of an affine stratification is the largest k such that ∪_{j}Y_{k,j} is nonempty. The affine stratification number asn X of a scheme X is the minimum of the length over all possible affine stratifications of X.

The inclusion in (1)refers to the underlying set. We do not require that each Y_{k,i} be irreducible. We also do not require any relation between k and the dimension or codimension of Y_{k,i} in X. We will see however (Theorem 3.1) that it is always possible to assume that the stratification has a very nice form.

Strictly speaking, the term “stratification” is inappropriate, as \overline{Y}_{k,i} \ Y_{k,i} need not be a union of Y_{k′,j}: let X be the co-ordinate axes in 𝔸^{2}, Y_{0,1} the x-axis minus the origin, and Y_{1,1} the y-axis. However, Theorem 3.1(a) shows that we may take (1) to be an actual stratification.

The affine stratification number has many good properties, including the following (Lemma 2.1, Propositions 4.2, 4.6, 2.10).

• asn X = 0 if and only if X is affine.

• asn X ≤ dim X.

• asn X ≤ acn X. (Equality does not always hold.)

• asn(X × Y) ≤ asn X + asn Y.

• If D is an effective Cartier divisor on X, then asn(X − D) ≤ asn X.

• If Y → X is an affine morphism, then asn Y ≤ asn X.

Even if one is only interested in equidimensional schemes, the more general Definition 1.2 has advantages over Definition 1.1. For example, the last property is immediate using Definition 1.2, but not obvious using Definition 1.1.

In Section 2, we establish basic properties of affine stratifications. In Section 3, we show that affine stratifications can be reorganized into a particularly good form. In particular, if X is equidimensional, then easn X = asn X (Proposition 3.7), so the notation easn may be discarded. In Section 4, we give topological consequences of bounded asn.

Our motivation is to bound the affine stratification number of moduli spaces (in particular, of pointed curves) to obtain topological and cohomological consequences. We describe our work in progress in the form of several conjectures in Section 5. For example, the conjectures bound the homotopy type of the moduli spaces of curves (a) of compact type (stable curves whose dual graph is a tree, or equivalently stable curves with compact Jacobian), (b) with “rational tails” (stable genus g curves having a smooth component of genus g), and (c) with at most k rational components (a locus introduced in [GV]), see Proposition 5.9.

1.3. Acknowledgments. This note arose from our ongoing efforts to prove a conjecture of E. Looijenga, and much of what is here derives from questions, ideas, and work of his. In particular, we suspect that he is aware of most of the results given here, and that we are following in his footsteps. We thank him for inspiration. We also thank J. Starr for Example 4.8, W. Fulton for pointing out the examples of Nori [N] and Horrocks [Ho], and T. Graber for helpful conversations. Finally, we are grateful to the organizers of the 2003 conference Algebraic structures and moduli spaces, at the Centre de Recherches Mathématiques (CRM), which led to this work.

2. Basic properties of affine stratifications

The most basic property is that an affine stratification always exists, and hence asn X is defined for any scheme X: if ∪_{i=0}^{n}U_{i} is a covering of X by open affine sets, then

(2)  U_{0} ⊔ (U_{1} \ U_{0}) ⊔ (U_{2} \ (U_{0} ∪ U_{1})) ⊔ ⋯

gives an affine stratification of X.

The following lemma is trivial.

2.1. Lemma. —

(a) The affine stratification number depends only on the reduced structure of X, i.e. asn X = asn X^{red}.

(b) If X → Y is an affine morphism, then asn X ≤ asn Y.

(c) asn(X × Y) ≤ asn X + asn Y.

(d) If D is an effective Cartier divisor on X, then asn(X − D) ≤ asn X.

Part (d) requires the following well-known fact.

2.2. Lemma. — Any irreducible affine scheme X, minus an effective Cartier divisor D, is affine.

(Reason: the inclusion X \ D ↪ X is an affine morphism, since this can be verified locally. But X is affine.)

Here is a partial converse to Lemma 2.2. A more precise converse is given in Proposition 2.6.

2.3. Lemma. — Suppose that V is an irreducible affine scheme, and that U ⊂ V is an open affine subset. Then the complement Z := V \ U is a Weil divisor in V.

Proof. We first assume that V (and hence U) is normal. Let Z = ∪_{i}Z_{i} be the decomposition of Z into irreducible components, and let Z′ = ∪_{j}Z_{j} be the union of those components of codimension one in V. We set U′ = V \ Z′, and let i : U ↪ U′ be the natural open immersion. Since U′ is normal, and the complement of U in U′ is of codimension at least 2 in U′, we have i_{∗}O_{U} = O_{U′}. We will use this and the fact that both U and V are affine to see that U = U′.

Let A = Γ(V, O_{V}) and B = Γ(U, O_{U}) = Γ(U′, O_{U′}). We have an inclusion of rings A ↪ B corresponding to the opposite inclusion of open sets. Suppose that U ≠ U′, and let x be any point of U′ \ U. Since V is affine, x corresponds to a prime ideal P_{x} of A. Since x ∈ U′, no element of P_{x} can be a unit in Γ(U′, O_{U′}), and hence P_{x} remains a prime ideal in B, which is a localization of A. Therefore, since U is affine, x ∈ U, contrary to assumption.

Passing to the general case, we drop the assumption that V and U are normal, and let Ṽ and Ũ be their normalizations. We have the commutative diagram

\xymatrix{Ũ \ar@{^{(}->}[r] \ar[d] & Ṽ \ar[d] \\ U \ar@{^{(}->}[r] & V}

where the vertical arrows are the normalization maps, and the horizontal arrows are open immersions. By the first part of the lemma, the complement Z̃ of Ũ in Ṽ is of codimension one in Ṽ. Since Z̃ maps finitely and surjectively onto Z, dim(Z) = dim(Z̃), and hence Z is of codimension one in V. □

The next corollary follows immediately. (Note that X need not be equidimensional here.)

2.4. Corollary. — The complement of a dense affine open subset in any scheme is of pure codimension one.

2.5. Examples. (a) Let X be the affine cone over an elliptic curve, embedded in ℂℙ^{2} as a cubic. Let Z be the cone over any point of the curve of infinite order in the group law. Then X \ Z is affine, but Z is not ℚ-Cartier. This shows that the complement of an affine open set in an affine scheme need not be the support of a Cartier divisor: we cannot hope to improve the conclusion of Lemma 2.3 to match the hypothesis of Lemma 2.2.

(b) Let S be ℙ^{2} blown up at a point, and let X be the affine cone over some projective embedding of S. Let Z ⊂ X be the affine cone over the exceptional divisor of the blowup. Then Z is of codimension one in X, but cd(X \ Z) = 1, so in particular it is not affine. This shows that, conversely, the complement of a Weil divisor in an affine scheme need not be affine: we can not hope to improve the hypothesis of Lemma 2.2 to match the conclusion of Lemma 2.3.

However, there is a more precise statement giving a necessary and sufficient condition on a closed subset Z of an affine scheme V for the complement V \ Z to be affine.

2.6. Proposition. — Let V be an affine scheme (possibly reducible) and Z a closed subset of V. Then U := V \ Z is affine if and only if H_{Z}^{i}(F) = 0 for all quasicoherent sheaves F on V and all i ≥ 2.

Here H_{Z}^{i}(F) is the local cohomology group. This also implies the same fact for the local cohomology sheaves H_{Z}^{i}(F), see Corollary 2.7(a) below.

Proof. Let F be any quasicoherent sheaf on V. We have the long exact excision sequence of cohomology groups

(3)  \begin{aligned}0 & ⟶ & H_{Z}^{0}(F) & ⟶ & H^{0}(V, F) & ⟶ & H^{0}(U, F|_{U}) \\ & ⟶ & H_{Z}^{1}(F) & ⟶ & H^{1}(V, F) & ⟶ & H^{1}(U, F|_{U}) \\ & ⟶ & H_{Z}^{2}(F) & ⟶ & H^{2}(V, F) & ⟶ & H^{2}(U, F|_{U}) & ⟶ & · · · . \end{aligned}

Since V is affine, we have H^{i}(V, F) = 0 for all i ≥ 1, so that H^{i}(U, F|_{U}) = H_{Z}^{i+1}(F) for all i ≥ 1. Hence (using Serre’s criterion for affineness) U is affine if and only if H_{Z}^{i}(F) = 0 for all i ≥ 2 and all quasicoherent sheaves F. □

2.7. Corollary. — Let X be a scheme (possibly reducible) and U a dense affine open subset. Let Z := X \ U. For any quasicoherent sheaf F on X,

(a) H_{Z}^{i}(F) = 0 for all i ≥ 2, and

(b) H_{Z}^{i}(F) = 0 for all i > cd Z + 1.

The notation cd denotes cohomological dimension, see Section 1.

Proof. (a) Since the local cohomology sheaf H_{Z}^{i}(F) is the sheafification of the functor V ↦ H_{Z∩V}^{i} (F|_{V}) [GH, Proposition 1.2], it is sufficient to check that the local cohomology group vanishes for sufficiently small V around any point of Z. But if V is any open affine set, then V ∩ U is nonempty (since U is dense) and also affine (by separatedness). Hence H_{Z∩V}^{i} (F|_{V}) = 0 for i ≥ 2 by Proposition 2.6 and so H_{Z}^{i}(F) = 0 as well.

(b) The local cohomology sheaves H_{Z}^{i}(F) are quasicoherent and are supported on Z. The local cohomology groups can be computed by a spectral sequence with E_{2}^{pq} term H^{p}(X, H_{Z}^{q}(F)) = H^{p}(Z, H_{Z}^{q}(F)). Since H^{p}(Z, ·) = 0 for p > cd Z, and H_{Z}^{q}(F) = 0 for q > 1 by part (a), we have H_{Z}^{i}(F) = 0 for i > cd Z + 1. □

2.8. Corollary. — Let X be a scheme, U a dense affine open subset, and set Z := X \ U. Then cd X ≤ cd Z + 1.

Proof. For any any quasicoherent sheaf F on X, the excision sequence (3) and the fact that U is affine gives H^{i}(X, F) = H_{Z}^{i}(F) for all i ≥ 2, and that H^{1}(X, F) is a quotient of H_{Z}^{1}(F). Hence, for any i ≥ 1, H_{Z}^{i}(F) = 0 implies that H^{i}(X, F) = 0. Since H_{Z}^{i}(F) = 0 for all i > cd Z + 1 by Corollary 2.7(b), we have cd X ≤ cd Z + 1. □

2.9. Bounding asn by finite flat covers. The following result is useful to bound asn X by studying covers of X.

2.10. Proposition. — Suppose π : Y → X is a surjective finite flat morphism of degree not divisible by the characteristic of the base field, and Y is affine. Then X is affine.

Proof. The hypothesis implies that π_{∗}O_{Y} is a coherent locally free sheaf on X. The trace map gives a splitting π_{∗}O_{Y} ≅ O_{X} ⊕E for some vector bundle E on X. If F is any coherent sheaf on X, then the flatness of π gives π_{∗}π^{∗}F = F ⊕ (E ⊗ F), and it then follows from the Leray spectral sequence and the finiteness of π that H^{i}(X, F) is a direct summand of H^{i}(Y, π^{∗}F) for all i ≥ 0. Since Y is affine, these vanish if i ≥ 1, hence the cohomology groups on X do as well, and therefore X is affine by Serre’s criterion for affineness. □

3. Reorganizing affine stratifications

We describe various ways that we can reorganize the stratification which are more convenient for analyzing X. The main results of this section are summarized in the following theorem.

3.1. Theorem. — If X is any scheme and asn X = m, then there exists an affine stratification {Z_{0}, …, Z_{m}} of X such that for any k ≤ m:

(a) \overline{Z}_{k} = ∪_{k′≥k}Z_{k′},

(b) each Z_{k} is a dense open affine subset of \overline{Z}_{k}, and

(c) \overline{Z}_{k} is of pure codimension one in \overline{Z}_{k−1} .

If in addition X is equidimensional, then we also have

(d) each \overline{Z}_{k′} is of pure codimension k′ − k in \overline{Z}_{k} for any k′ ≥ k.

Even if X is not equidimensional, if we have an affine stratification {Y′_{k,l}} of length M such that each Y′_{k,i} is of pure codimension k in X, then setting Z′_{k} := ∪_{i}Y′_{k,i} for k = 0, …, M we have

(e) the affine stratification {Z′_{0}, …, Z′_{M}} satisfies (a)–(d) above.

(We are not guaranteed that M = m, so this stratification may not be optimal.)

The proof is summarized in Section 3.9. In analogy with CW-complexes, we define an affine cell decomposition of a scheme X to be an affine stratification

X = ⨆_{k} Z_{k}

where the Z_{k}’s satisfy (a)–(c) of Theorem 3.1. The theorem guarantees that such a decomposition exists for any scheme X, with length asn X.

3.2. Lemma. — Let {Y_{k,i}} be an affine stratification of a scheme X and let Z_{k} := ∪_{i}Y_{k,i} be the union of all the affine pieces of index k. Then each Z_{k} is an open dense affine subset of \overline{Z}_{k}, i.e. Z_{k} is locally closed and affine.

Proof. By definition, Z_{k} is a dense subset of \overline{Z}_{k}. We will see that it is an open subset, and most importantly, affine.

Since the affine stratification is finite, we have \overline{∪_{i}Y_{k,i}} = ∪_{i}\overline{Y}_{k,i}. For any distinct Y_{k,i} and Y_{k,j} and any point y ∈ \overline{Y}_{k,i} ∩ \overline{Y}_{k,j}, the fact that the Y ’s are disjoint, along with the stratification condition (1), implies that y must be in some Z_{k′} with k′ > k. In particular, y is in neither Y_{k,i} nor Y_{k,j}.

If we let C_{i} := ∪_{j≠i}\overline{Y}_{k,j} be the closed subset consisting of the closures of other Y_{k,j}’s, and V_{i} := X \ C_{i} the open complement, then the previous remark shows that Y_{k,i} ⊆ V_{i}, and therefore that Z_{k} ∩ V_{i} = Y_{k,i}.

Since every locally closed subset is an open subset of its closure, Y_{k,i} is an open subset of \overline{Y}_{k,i} ∩ V_{i} = \overline{Z}_{k} ∩ V_{i}. Since \overline{Z}_{k} ∩ V_{i} is an open subset of \overline{Z}_{k}, we see that Y_{k,i} is an open subset of \overline{Z}_{k}, and therefore that Z_{k} = ∪_{i}Y_{k,i} is an open subset of \overline{Z}_{k}.

Let Z̃_{k} be the disjoint union

Z̃_{k} = ⨆_{i} Y_{k,i},

and f : Z̃_{k} ⟶ X the natural morphism with image Z_{k}. The map f is one-to-one on points, and the fact that Z_{k} ∩ V_{i} = Y_{i,k} for each i implies that f is a homeomorphism, and in fact an immersion. Therefore, Z̃_{k} ≅ Z_{k} as schemes, and so Z_{k} is affine since Z̃_{k} is. □

3.3. Proposition. — Let {Y_{k,i}} be an affine stratification of X of length m. Then there exists an affine stratification {Y′_{k,j}} of length at most m such that the generic points of all components of X are contained in the zero stratum ∪_{j}Y′_{0,j} of {Y′_{k,j}}.

Proof. We first set Y′_{0,i} := Y_{0,i} for all valid indices i. Now let Y_{k,i} be any piece of the stratification with k ≥ 1. If Y_{k,i} does not contain the generic point of any component of X then set Y′_{k,i} := Y_{k,i}. On the other hand, suppose that Y_{k,i} contains η_{1}, …, η_{r} where each η_{j} is a generic point of X. In this case, for each j ∈ {1, …, r} choose an open affine subset U_{j} of Y_{k,i} containing η_{j} so that U_{j} intersects no components of X other than \overline{\{η_{j}\}}. Now set Y′_{k,j} := Y_{k,j} \ (∪_{j=1}^{r}U_{j}), and add the U_{j} in as elements of the zero stratum, Y′_{0,i_{j}} := U_{j}, where the i_{j} are chosen not to conflict with previously existing indices. It is straightforward to verify that this decomposition satisfies the affine stratification condition (1). □

3.4. Lemma. — Let {Y_{k,i}} be an affine stratification of X of length m. Then there exists an affine stratification {Y′_{k,i}} of length at most m such that if Z′_{k} := ∪_{i}Y′_{k,i} is the union of all affine pieces of index k, then for any k,

\overline{Z}′_{k} = ∪_{k′≥k}Z′_{k′}.

Proof. By Proposition 3.3 we may assume that all the generic points of components of X occur in the zero stratum of \{Y_{k,i}\}, and therefore that ∪_{i}\overline{Y}_{0,i} = X. We now proceed by induction on the length m of the stratification, the case m = 0 being trivial.

Let U := ∪_{i}Y_{0,i} be the union of the pieces in the zero stratum, and Z := ∪_{k≥1,i}Y_{k,i} the complement. Note that U is open and hence Z is closed by Lemma 3.2.

The {Y_{k,i}} with k ≥ 1 form an affine stratification of Z of length m − 1 (after reindexing the k’s to start with zero). Therefore by induction Z has an affine stratification of length at most m − 1 satisfying the hypothesis of the lemma. Reindexing the k’s again, and adding the Y_{0,i}’s as the zero stratum, we end up with an affine stratification {Y′_{k,i}} of length at most m which also satisfies the hypothesis of the lemma, completing the inductive step. □

3.5. Corollary. — For any scheme X, if asn X = m then there is an affine stratification \{Z_{0}, …, Z_{m}\} of X with \overline{Z}_{k} = ∪_{k′≥k}Z_{k′} for each k, and such that each Z_{k} is an open dense affine subset of \overline{Z}_{k}.

Proof. Combine Lemmas 3.2 and 3.4. □

3.6. Corollary. — For any scheme X, asn X ≤ dim X.

Proof. Let m = asn X and {Z_{0}, …, Z_{m}} be a stratification as in Corollary 3.5. By Corollary 2.4, each \overline{Z}_{k+1} is of pure codimension one in \overline{Z}_{k}. If \overline{Z}′_{m} is any irreducible component of \overline{Z}_{m}, then that means we can inductively find a chain of closed irreducible subsets \overline{Z}′_{m} ⊂ \overline{Z}′_{m−1} ⊂ \overline{Z}′_{m−2} ⊂ ⋯ ⊂ \overline{Z}′_{1} ⊂ \overline{Z}′_{0}, with each \overline{Z}′_{k} an irreducible component of \overline{Z}_{k}. Then dim X ≥ dim \overline{Z}′_{m} + m ≥ m. □

If we assume an additional hypothesis about X or the stratification {Y_{k,i}}, we have slightly stronger results about the stratification {Z_{0}, …, Z_{m}} of Corollary 3.5.

3.7. Proposition. — If X is an equidimensional scheme, and {Z_{0}, …, Z_{m}} the stratification of Corollary 3.5, then we have in addition that \overline{Z}_{k′} is of pure codimension k′ − k in \overline{Z}_{k} for all k′ ≥ k. In particular, easn X = asn X.

Proof. By Corollary 2.4 each \overline{Z}_{k+1} is of pure codimension one in \overline{Z}_{k}. If \overline{Z}_{0} = X is equidimensional, then it follows that each \overline{Z}_{k} is equidimensional as well, and from this that \overline{Z}_{k′} is of pure codimension k′ − k in \overline{Z}_{k} for any k′ ≥ k. □

Even if X is not equidimensional, if the affine stratification {Y_{k,i}} satisfies a suitable condition we get a similar good result about the stratification by the Z_{k}’s.

3.8. Proposition. — Let {Y_{k,i}} be an affine stratification of a scheme X with each Y_{k,i} of pure codimension k in X. Let Z_{k} := ∪_{i}Y_{k,i} be the union of all the affine pieces of codimension k. Then

(i) For k′ ≥ k, \overline{Z}_{k′} is of pure codimension k′ − k in \overline{Z}_{k}; in particular, \overline{Z}_{k′} ⊆ \overline{Z}_{k}.

(ii) \overline{Z}_{k} = ∪_{k′≥k}Z_{k′}.

Proof. Since the decomposition is finite, the irreducible components of \overline{Z}_{m} are all of the form \overline{W}_{m} with W_{m} an irreducible component of some Y_{m,j}.

We prove (i) by induction on k. For k = 0 the result is obvious, since \overline{Z}_{0} = X, and \overline{Z}_{k′} is of pure codimension k′ in X. So assume that k > 0 and that (i) is true for k − 1.

Let \overline{W}_{k′} by any irreducible component of \overline{Z}_{k′} with k′ ≥ k. By the induction hypothesis, \overline{W}_{k′} ⊂ \overline{Z}_{k−1}, and is of codimension k′ − k + 1 in \overline{Z}_{k−1}. Let T_{k−1} be any irreducible component of Z_{k−1} whose closure contains \overline{W}_{k′} and such that codim(\overline{W}_{k′}, \overline{T}_{k−1}) = k′ − k + 1. Lemma 3.2 gives us that Z_{k−1} is affine, and therefore T_{k−1} is affine also.

By Lemma 2.3, the closed set \overline{T}_{k−1} \ T_{k−1} has codimension one in \overline{T}_{k−1}. Let η_{k} be the generic point of any component of \overline{T}_{k−1} \ T_{k−1} containing W_{k′}; one exists by our choice of T_{k−1}. Since codim(\overline{W}_{k′}, \overline{T}_{k−1}) = k′ − k + 1 and codim(\overline{\{η_{k}\}}, \overline{T}_{k−1}) = 1, we have codim(W_{k′}, \overline{\{η_{k}\}}) = k′ − k.

The Z_{m}’s partition X, and so η_{k} must be in exactly one Z_{m}. We cannot have m ≤ k − 1, since that would contradict the stratification condition. We cannot have m ≥ k + 1, since this would contradict codim(\overline{Z}_{m}, \overline{Z}_{k−1}) = m − k + 1, which holds by the induction hypothesis. Therefore η_{k} is in Z_{k}, and so \overline{W}_{k−1} ⊂ \overline{Z}_{k}.

We have already seen that codim(W_{k′}, \overline{\{η_{k}\}}) = k′ − k. Since

codim (\overline{W}_{k′}, \overline{Z}_{k}) = sup_{i} (codim (\overline{W}_{k′}, \overline{Y}_{k,i})) ,

with \overline{Y}_{k,i} running over the components of \overline{Z}_{k}, we get codim(\overline{W}_{k′}, \overline{Z}_{k}) ≥ k′ − k. But for any three closed schemes \overline{W}, \overline{Z}, and X with \overline{W} ⊆ \overline{Z} ⊆ X, we always have

codim (\overline{W} , \overline{Z}) + codim (\overline{Z}, X) ≤ codim (\overline{W} , X) .

Since the codimensions of \overline{W}_{k′} in X, and \overline{Z}_{k} in X are k′ and k by hypothesis, this gives codim(\overline{W}_{k′}, \overline{Z}_{k}) ≤ k′ − k, and hence codim(\overline{W}_{k′}, \overline{Z}_{k}) = k′ − k. Therefore \overline{Z}_{k′} is contained in \overline{Z}_{k}, and is of pure codimension k′ − k, completing the inductive step for (i).

To prove (ii), the stratification condition gives \overline{Z}_{k} ⊆ ∪_{k′≥k}Z_{k′}, while part (i) above gives the opposite inclusion. □

3.9. Proof of Theorem 3.1. Part (a) is Lemma 3.4, part (b) Lemma 3.2, part (c) Corollary 2.4, part (d) Proposition 3.7, and part (e) Proposition 3.8 and Lemma 3.2 again. □

4. Topological consequences of bounded affine stratification number

We now describe the topological consequences of bounded asn, in particular: relation to dimension (Section 4.1), affine covering number (Section 4.5), cohomological dimension (for quasicoherent sheaves, Section 4.11, as well as constructible and ℓ-adic sheaves, Section 4.15), dimension of largest proper subscheme (Section 4.14), and homotopy type (Section 4.20).

4.1. Relation to dimension.

4.2. Proposition. — asn X ≤ dim X. If one top dimensional component of X is proper, then equality holds.

The first statement is Corollary 3.6. The second statement follows from Proposition 4.12 (cd ≤ asn) and the following theorem, first conjectured by Lichtenbaum.

4.3. Theorem (Grothendieck [GH, 6.9], Kleiman [K, Main Theorem]). — If d = dim X, then cd X = d if and only if at least one d-dimensional component of X is proper.

4.4. Example: All values between 0 and dim X are possible. Let X_{k} = ℙ^{n}\{(n − k − 1)-plane)}, for k between 0 and n−1. Then clearly cd X_{k} = k and asn X_{k} ≤ k. We will see that cd ≤ asn (Proposition 4.12), from which the result follows.

4.5. Relation to affine covering number. Recall that the affine covering number acn X of a scheme X is the minimal number of affine open subsets required to cover X, minus 1. The invariant acn does not obviously behave as well as asn with respect to products (cf. Lemma 2.1(c)); it also is not bounded by dimension (Example 4.8 below).

The argument of (2)gives the following.

4.6. Proposition. — asn X ≤ acn X.

4.7. Example. In general, acn X ≠ asn X. As an example, let X be a complex K3 surface with Picard rank 1, minus a very general point. Then acn X = 2: if for every point p of X, acn(X − p) = 1, then (given the hypothesis that the Picard rank is 1) any two points of X are equivalent in A_{0}(Y) (with ℚ-coefficients), contradicting Mumford’s theorem that A_{0}(Y) is not countably generated [M]. Example 4.8 below gives another example (in light of Proposition 4.2).

4.8. Example: acn X may be larger than dim X. When X is quasiprojective, acn X ≤ dim X. (Reason: Let \overline{X} be a projective compactification such that the complement \overline{X}\X is a Cartier divisor D. Consider an embedding \overline{X} ↪ ℙ^{n} and let H_{0}, …, H_{dim X} be hypersurfaces so that \overline{X} ∩ H_{0} ∩ ⋯ ∩ H_{dim X} = ∅. Then the \overline{U_{i}} := \overline{X} \ (\overline{X} ∩ H_{i}) form an affine cover of \overline{X}. We conclude using Lemma 2.2.)

However, the following example, due to J. Starr, shows that acn X may be greater than dim X. Given any n, we describe a reducible, reduced threefold that requires at least n affine open sets to cover it. Recall Hironaka’s example (e.g. [Ha, Example B.3.4.1]) of a nonsingular proper nonprojective threefold X. Nonprojectivity is shown by exhibiting two curves ℓ and m whose sum is numerically trivial. Hence no affine open set can meet both ℓ and m; otherwise its complement would be a divisor (Lemma 2.3), hence Cartier (as X is nonsingular), which meets both ℓ and m positively. Now choose points p and ℓ and q on m. Consider \binom{n}{2} copies of (X, p, q), corresponding to ordered pairs (i, j) (1 ≤ i < j ≤ n); call these copies (X_{ij}, p_{ij}, q_{ij}). Let r_{1}, …, r_{n} be copies of a reduced point. Glue r_{i} to p_{ji} and q_{ik}. Then no affine open can contain both r_{i} and r_{j} for i < j (by considering X_{ij}).

4.9. Example: a family of integral threefolds with arbitrary high affine covering number (and no smooth embeddings). This leads to an example of an integral (but singular) threefold that requires at least n affine open sets to cover it. (Question: Is there a family of nonsingular irreducible varieties of fixed dimension with unbounded affine covering number?) Our example will be a blow-up of ℙ^{3}. Choose n curves C_{1}, …, C_{n} in ℙ^{2} ⊂ ℙ^{3} that meet in n simple n-fold points p_{1}, …, p_{n} (and possibly elsewhere). Away from p_{1}, …, p_{n} blow up C_{1}, …, C_{n} in some arbitrary order. In a neighborhood of p_{i} (not containing any other intersection of the C_{j}) blow up C_{i} first (giving a smooth threefold) and then blow up the local complete intersection ∪_{j≠i}C_{j} (or more precisely, the proper transform thereof), giving a threefold with a single singularity (call it q_{i}). The preimage of p_{i} is the union of two ℙ^{1}’s, one arising from the exceptional divisor of C_{i} (call it ℓ_{i}), and one from the exceptional divisor of ∪_{j≠i}C_{j}; they meet at q_{i}. By Hironaka’s argument, ℓ_{i} +ℓ_{j} is numerically trivial for all i ≠ j. Then no affine open U can contain both q_{i} and q_{j}: the complement of U would be a divisor, meeting ℓ_{i} and ℓ_{j} properly and at smooth points of our threefold (i.e. not at q_{i} and q_{j}), and the same contradiction applies.

This is also an example of a scheme which cannot be embedded in any smooth scheme, or indeed algebraic space. (Earlier examples are the topic of papers of Horrocks [Ho] and Nori [N].) If X ↪ W with W smooth, then for any divisor D (automatically Cartier) on W, D · ℓ_{i} + D · ℓ_{j} = 0 for all i, j. If n ≥ 3, this implies that D · ℓ_{i} = 0 for all i. But for any affine open set U of W with U ∩ ℓ_{i} ≠ ∅, the complement D = W \ U would intersect ℓ_{i} properly, giving us the contradiction D · ℓ_{i} > 0. Hence no such embedding is possible.

By combining Proposition 4.2 with Proposition 4.6, we obtain the following.

4.10. Proposition. — If X is proper, then acn X ≥ dim X.

We note that this also follows from Theorem 4.3. Example 4.7 shows that it is not true that acn X = dim X if and only if X is proper, even for quasiprojective X.

4.11. Relation to cohomological dimension. Just as the dimensions of the cells in a CWcomplex bounds the topological (co)homology, the length of the stratification into affine cells bounds the quasicoherent sheaf cohomology.

4.12. Proposition. — cd X ≤ asn X.

Proof. We prove the result by induction on asn X. It is clear for asn X = 0, so assume that m := asn X > 0 and that the result is proven for all schemes Z with asn Z < asn X. Let {Z_{0}, …, Z_{m}} be an affine cell decomposition given by Theorem 3.1. Set Z := X \ Z_{0} = \overline{Z}_{1} = ∪_{k≥1}Z_{k}.

By Theorem 3.1(b) Z_{0} is an open dense affine subset of X, so by Corollary 2.8, cd X ≤ cd Z+1. Next, Z = ⊔_{k=1}^{m−1}Z_{k} is (after reindexing) an affine stratification of Z of length m−1, so asn Z ≤ asn X − 1. Finally, by the inductive hypothesis, cd Z ≤ asn Z. Combining these three inequalities gives cd X ≤ asn X, completing the inductive step. □

We remark in passing that by combining Proposition 4.12 with Corollary 3.6 we obtain another proof of Grothendieck’s dimensional vanishing theorem ([G, Theorem 3.6.5], [Ha, Theorem III.2.7]).

We conclude with an obvious result.

4.13. Proposition. — cd X = 0 if and only if asn X = 0 if and only if acn X = 0.

Proof. Each of the three is true if and only if X is affine (the first by Serre’s criterion for affineness). □

4.14. Relation to dimension of largest complete subscheme. Motivated by Diaz’ theorem [D], let psv X be the largest dimension of a proper closed subscheme of X. If Z is a proper closed subscheme of X (with inclusion j : Z ↪ X), and if F is a quasicoherent sheaf on Z, then j_{∗}F is a quasicoherent sheaf on X, and H^{i}(X, j_{∗}F) = H^{i}(Z, F) for all i. By Theorem 4.3 we can find a quasicoherent sheaf F on Z with H^{dim Z}(Z, F) ≠ 0, and so this gives psv X ≤ cd X. Hence by Proposition 4.12,

psv X ≤ asn X.

4.15. Relation to cohomological vanishing for constructible and ℓ-adic sheaves. In this section all notions related to sheaves (including stalks, pushforwards, and cohomology groups) are with respect to the étale topology. For instance, “sheaf on X” means “sheaf on X in the étale topology”.

To show how asn implies cohomological vanishing for constructible and ℓ-adic sheaves (Corollary 4.19), we first recall a theorem and some notation of Artin. For any (étale) sheaf F of abelian groups on X, let

d(F) := sup \{dim (\overline{\{x\}}) | x ∈ X, F\overline{x} ≠ 0\}

be the dimension of the support of F.

4.16. Artin’s Theorem [A2, Theorem 3.1]. — Let f : X ⟶ Y be an affine morphism of schemes of finite type over a field k, and F a torsion sheaf (i.e. sheaf of torsion groups) on X. Then d(R^{q}f_{∗}F) ≤ d(F) − q for all q ≥ 0.

We will apply Artin’s Theorem in the following form:

4.17. Proposition. — Suppose that X is a scheme, U an affine open subset of X, and Z := X \ U the complement. Then for any torsion sheaf F on X,

d(H_{ét,Z}^{q}(F)) ≤ d(F) − q + 1.

Here the H_{ét,Z}^{q}(F) are the local cohomology sheaves in the étale topology. The usual excision and spectral sequences for local cohomology remain true in the étale setting, see [V, Sec. 6].

Proof. If i : U ↪ X is the inclusion, then for any sheaf F of abelian groups on X we have the exact sequence [V, Proposition 6.5]

0 ⟶ H_{ét,Z}^{0}(F) ⟶ F ⟶ i_{∗}(F|_{U}) ⟶ H_{ét,Z}^{1}(F) ⟶ 0,

as well as isomorphisms

H_{ét,Z}^{q}(F) ≅ R^{q−1}i_{∗}(F|_{U}) for all q ≥ 2.

If q ≥ 2 the proposition then follows from the above isomorphism and Artin’s Theorem 4.16 applied to the inclusion morphism i, which is affine since U is.

If q = 1 it is enough to bound d(i_{∗}(F|_{U})), since H_{ét,Z}^{1}(F) is a quotient of i_{∗}(F|_{U}). The points x ∈ X where (i_{∗}(F|_{U}))\overline{x} ≠ 0 are the points x ∈ U with F\overline{x} ≠ 0 and points x ∈ Z such that there exists a point x′ ∈ U, x ∈ \overline{\{x′\}} with F\overline{x′} ≠ 0. In particular, the support of i_{∗}(F|_{U}) is contained in the support of F, so d(H_{ét,Z}^{1}(F)) ≤ d(F), which is exactly the statement of the proposition when q = 1.

If q = 0 note that d(H_{ét,Z}^{0}(F)) ≤ d(F) since H_{ét,Z}^{0}(F) is a subsheaf of F, while the proposition only claims the weaker bound d(H_{ét,Z}^{0}(F)) ≤ d(F) + 1. □

4.18. Lemma. — If F is a torsion sheaf on X, then Hét^{n}(X, F) = 0 for all n > d(F) + asn X.

Proof. We show the result by induction on asn X, the case asn X = 0 being Artin’s Theorem 4.16 again. Let {Z_{0}, …, Z_{asn X}} be an affine cell decomposition of X (as given by Theorem 3.1). Set Z := X \ Z_{0} = ∪_{k≥1}Z_{k}.

We have Hét^{n}(Z_{0}, F|_{Z_{0}}) = 0 for all n > d(F|_{Z_{0}}) by Artin’s Theorem, and since d(F|_{Z_{0}}) ≤ d(F) the excision sequence (equation (3) holds in this context, [V, (6.5.3)]) shows that Hét^{n}(X, F) is a quotient of H_{ét,Z}^{n}(F) for all n > d(F). It is therefore enough to show that H_{ét,Z}^{n}(F) = 0 for n > d(F) + asn X.

We can compute H_{ét,Z}^{n}(F) by a spectral sequence with E_{2}^{pq} term Hét^{p}(X, H_{ét,Z}^{q}(F)) ([V, Proposition 6.4]). We have Hét^{p}(X, H_{ét,Z}^{q}(F)) = Hét^{p}(Z, H_{ét,Z}^{q}(F)) since H_{ét,Z}^{q}(F) is supported on Z. By Proposition 4.17 we have d(H_{ét,Z}^{q}(F)) ≤ d(F) − q + 1. Since asn Z < asn X we can apply the inductive hypothesis to conclude that Hét^{p}(Z, H_{ét,Z}^{q}(F)) = 0 for p > d(F) − q + 1 + asn Z, or p + q > d(F) + asn Z + 1. Again using asn Z < asn X, this gives H_{ét,Z}^{n}(F) = 0 for n > d(F) + asn X. □

4.19. Corollary. —

(a) If F is a torsion sheaf, then Hét^{n}(X, F) = 0 for all n > dim X + asn X.

(b) If F is a constructible sheaf, then Hét^{n}(X, F) = 0 for all n > dim X + asn X.

(c) If F_{ℓ} is an ℓ-adic sheaf on X, then Hét^{n}(X, F_{ℓ}) = 0 for all n > dim X + asn X.

Proof. (a) Clearly we have d(F) ≤ dim X. (b) A constructible sheaf is a special case of a torsion sheaf (compare [A1, Proposition 1.2(ii)] with [A1, Def. 2.3]). (c) follows from (a). □

4.20. Relation to homotopy type. We expect that the affine stratification number bounds the homotopy type as follows.

4.21. Conjecture. — If the base field is ℂ, then X has the homotopy type of a finite complex of dimension at most asn X + dim X.

5. Applications to moduli spaces of curves

One motivation for the definition of affine stratification number is the study of the moduli space of curves, and certain geometrically important open subsets. We will use Definition 1.1 (which we may, by Proposition 3.7).

5.1. Preliminary aside: the affine stratification number of Deligne-Mumford stacks. As we have only defined the affine stratification number of schemes, throughout this section, we will work with coarse moduli space of curves. One should presumably work instead with a more general definition for Deligne-Mumford stacks. One possible definition is to replace the notion of “affine” in the definition of affine stratification number with that of a Deligne-Mumford stack that has a surjective finite flat cover by an affine scheme (see Proposition 2.10).

5.2. Recall the following question of Looijenga’s.

5.3. Conjecture (Looijenga). — (a) acn M_{g} ≤ g − 2 for g ≥ 2. (b) More generally, acn M_{g,n} ≤ g − 1 − δ_{n,0} whenever g > 0, (g, n) ≠ (1, 0).

The case n = 1 of (b) implies the cases n > 1, as the morphism M_{g,n+1} → M_{g,n} is affine for n ≥ 1.

This suggests the following, weaker conjecture, which is straightforward to verify for small (g, n) (using Proposition 2.10 judiciously). We are currently pursuing a program to prove this (work in progress).

5.4. Conjecture (Looijenga [HL, p. 112, Problem 6.5]). — asn M_{g} ≤ g − 2 for g ≥ 2.

From this statement (and properties of asn), we obtain a number of consequences.

5.5. Proposition (Looijenga [HL, p. 112]). — Conjecture 5.4 implies that asn M_{g,n} ≤ g − 1 − δ_{n,0} whenever g > 0, (g, n) ≠ (1, 0).

Proof. As M_{g,n+1} → M_{g,n} is affine for n ≥ 1, it suffices to prove the result for M_{0,3} and M_{g,1} with g > 0. The cases g = 0 and g = 1 are immediate. For g > 1, let D be a multisection of M_{g,1} → M_{g} (e.g. a suitable Weierstrass divisor). Then the morphisms D → M_{g} and (M_{g,1} \ D) → M_{g} are affine and surjective, so pulling back the affine stratification of M_{g} to M_{g,1} and intersecting with (M_{g,1}\D)⊔D yields the desired affine stratification of M_{g,1}. □

Examination of small genus cases suggests the following refinement of Conjecture 5.4.

5.6. Conjecture. — There is an affine stratification of M_{g′,n′} preserved by the symmetric group acting on the n′ points. The induced decomposition of \overline{M}_{g,n} is a stratification.

This leads to a bound on the affine stratification number of the open subset \overline{M}_{g,n}^{≤k}, corresponding to stable n-pointed genus g curves with at most k genus 0 components, defined in [GV, Section 4].

5.7. Proposition. — Conjecture 5.6 implies that asn \overline{M}_{g,n}^{≤k} ≤ g − 1 + k for all g > 0, n > 0.

This is more evidence of the relevance of this strange filtration of the moduli space of curves. In particular, compare this to Theorem ⋆ of [GV], that the tautological ring of \overline{M}_{g,n}^{≤k} vanishes in codimension greater than g − 1 + k. (In [GV, p. 3], Looijenga asks precisely this question, with asn replaced by acn.)

Proof. We show that dimension of any stratum of \overline{M}_{g,n} appearing in \overline{M}_{g,n}^{≤k} is at least

3g − 3 + n − (g − 1 + k) = 2g − 2 + n − k.

This is true for strata in M_{g,n} by Proposition 5.5. Consider any other boundary stratum, say with j rational components (j ≤ k) with m_{1}, …, m_{j} special points respectively; and s other components, with genus g_{1}, …, g_{s} and n_{1}, …, n_{s} special points respectively. By Proposition 2.10, it suffices to pass to the finite étale cover that is isomorphic to

∏_{i=1}^{j} M_{0,m_{i}} × ∏_{i=1}^{s} M_{g_{i},n_{i}}.

By Proposition 5.5 (and using M_{0,m_{i}} affine), we can decompose this space into affine sets of dimension at least

∑_{i=1}^{j}(m_{i} − 3) + ∑_{i=1}^{s}(3g_{i} − 3 + n_{i} − (g_{i} − 1)).

Now ∑_{i=1}^{j}(m_{i} − 2) + ∑_{i=1}^{s}(2g_{i} − 2 + n_{i}) = 2g − 2 + n, so each affine set has dimension at least

2g − 2 + n − j

and thus codimension in \overline{M}_{g,n}^{≥k} at most g − 1 + j ≤ g − 1 + k as desired. □

This leads to bounds on other spaces of interest. Let M_{g,n}^{ct} be the open subset of \overline{M}_{g,n} corresponding to curves of compact type (i.e. with compact Jacobian, or equivalently with dual graph containing no loops). Let M_{g,n}^{rt} be the open subset corresponding to curves with rational tails (i.e. with a component a smooth genus g curve, or equivalently with dual graph with a genus g vertex).

5.8. Corollary. — Conjecture 5.6 implies that asn M_{g,n}^{ct} ≤ 2g − 3 + n and asn M_{g,n}^{rt} ≤ g + n − 2 for g > 0, n > 0.

Proof. M_{g,n}^{ct} is obtained by removing boundary strata from \overline{M}_{g,n}^{≤g+n−2}. M_{g,n}^{rt} is obtained by removing boundary strata from \overline{M}_{g,n}^{≤n−1}. □

5.9. Corollary. — Conjectures 4.21 and 5.6 imply that \overline{M}_{g,n}^{≤k} (resp. M_{g,n}^{ct}, M_{g,n}^{rt}) has the homotopy type of a finite complex of dimension at most 4g − 4 + n + k (resp. 5g − 6 + 2n, 4g − 5 + 2n).

References

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[A2] M. Artin, Théorème de finitude pour un morphisme propre; dimension cohomologique des schémas algébriques affines in SGA 4, t. 3, Exposé XIV. Lecture Notes in Math. 305, Springer-Verlag, New York, 1973, 145–167.

[D] S. Diaz, Complete subvarieties of the moduli space of smooth curves, in Algebraic geometry — Bowdoin 1985 (S. Bloch, ed.), Proc. Sympos. Pure Math. 46, Part 1, Amer. Math. Soc., Providence, RI, 1987, 77–81.

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[Ha] R. Hartshorne, Algebraic Geometry, Lecture Notes in Math. 52, Springer-Verlag, New York, 1977.

[HL] R. Hain and E. Looijenga, Mapping class groups and moduli spaces of curves, in Algebraic Geometry — Santa Cruz 1995, Proc. Sympos. Pure Math. 62, Part 2, Amer. Math. Soc., Providence, RI, 1997, 97–142.

[Ho] G. Horrocks, Birationally ruled surfaces without embeddings in regular schemes, Bull. London Math. Soc. 3 (1971), 57–60.

[K] S. Kleiman, On the vanishing of H^{n}(X, F) for an n-dimensional variety, Proc. Amer. Math. Soc. 18 (1967), 940-944.

[M] D. Mumford, Rational equivalences of 0-cycles on surfaces, J. Math. Kyoto Univ. 9 (1968), 195–204.

[N] M. Nori, Varieties with no smooth embeddings, in C.P. Ramanujam — a tribute, Tata Inst. Fund. Res. Studies in Math. 8, Springer, New York, 1978, 241–246.

[V] J. L. Verdier, Cohomologie dans les topos in SGA 4 t. 2, Exposé V. Lecture Notes in Math. 270, Springer-Verlag, New York, 1972, 1–80.

Dept. of Mathematics and Statistics, Queens University, Kingston, Ontario, Canada

  E-mail address: mikeroth@mast.queensu.ca

Dept. of Mathematics, Stanford University, Stanford, CA, USA

  E-mail address: vakil@math.stanford.edu