ESSENTIAL DIMENSION OF MODULI OF CURVES AND OTHER ALGEBRAIC STACKS

PATRICK BROSNAN^{†}, ZINOVY REICHSTEIN^{†}, AND ANGELO VISTOLI^{‡}

Abstract. In this paper we address questions of the following type. Let k be a base field and K/k be a field extension. Given a geometric object X over a field K (e.g. a smooth curve of genus g) what is the least transcendence degree of a field of definition of X over the base field k? In other words, how many independent parameters are needed to define X? To study these questions we introduce a notion of essential dimension for an algebraic stack. Using the resulting theory, we give a complete answer to the question above when the geometric objects X are smooth or stable curves.

Contents

1. Introduction 2 Acknowledgments 3 2. The essential dimension of a stack 3 3. A fiber dimension theorem 6 4. The essential dimension of a gerbe over a field 8 5. Gerbes over complete discrete valuation rings 11 5.1. Big and small étale sites 11 5.2. Group extensions and gerbes 12 5.3. Splitting the inertia sequence 13 5.4. Tame gerbes and splittings 15 5.5. Genericity 16 6. A genericity theorem for a smooth Deligne–Mumford stack 18 7. The essential dimension of M_{g,n} for (g, n) ≠ (1, 0) 21 8. Tate curves and the essential dimension of M_{1,0} 23 9. On the essential dimension of the stack

of principally polarized abelian varieties 25 References 29

2000 Mathematics Subject Classification. Primary 14A20, 14H10, 11E04. †Supported in part by an NSERC discovery grant. ‡Supported in part by the PRIN Project “Geometria sulle varietà algebriche”, financed

by MIUR.

1. Introduction

This paper was motivated by the following question.

Question 1.1. Let k be a field and g ≥ 0 be an integer. What is the smallest integer d such that for every field K/k, every smooth curve X of genus g defined over K descends to a subfield k ⊂ K_{0} ⊂ K with tr deg_{k} K_{0} ≤ d?

Here by “X descends to K_{0}” we mean that the exists a curve X_{0} over K_{0} such that X is K-isomorphic to X_{0} ×_{Spec K_{0}} Spec K.

In order to address this and related questions, we will introduce and study the notion of essential dimension for algebraic stacks; see §2. The essential dimension ed X of a scheme X is simply the dimension of X; on the other hand, the essential dimension of the classifying stack B_{k}G of an algebraic group G is the essential dimension of G in the usual sense; see [Rei00] or [BF03]. The notion of essential dimension of a stack is meant to bridge these two examples. The minimal integer d in Question 1.1 is the essential dimension of the moduli stack of smooth curves M_{g}. We show that ed X is finite for a broad class of algebraic stacks of finite type over a field; see Corollary 3.4. This class includes all Deligne–Mumford stacks and all quotient stacks of the form X = [X/G], where G a linear algebraic group.

Our main result is the following theorem.

Theorem 1.2. Let M_{g,n} (respectively, \overline{M}_{g,n}) be the stacks of n-pointed smooth (respectively, stable) algebraic curves of genus g over a field k of characteristic 0. Then

ed M_{g,n} = \begin{cases}2 & if (g, n) = (0, 0) or (1, 1); \\ 0 & if (g, n) = (0, 1) or (0, 2); \\ +∞ & if (g, n) = (1, 0); \\ 5 & if (g, n) = (2, 0); \\ 3g − 3 + n & otherwise. \end{cases}

Moreover for 2g − 2 + n > 0 we have ed \overline{M}_{g,n} = ed M_{g,n}.

In particular, the values of ed M_{g,0} = ed M_{g} give a complete answer to Question 1.1.

Note that 3g − 3 + n is the dimension of the moduli space M_{g,n} in the stable range 2g − 2 + n > 0 (and the dimension of the stack in all cases); the dimension of the moduli space represents an obvious lower bound for the essential dimension of a stack. The first four cases are precisely the ones where a generic object in M_{g,n} has non-trivial automorphisms, and the case (g, n) = (1, 0) is the only one where the automorphism group scheme of an object of M_{g,n} is not affine.

Our proof of Theorem 1.2 for (g, n) ≠ (1, 0) relies on two results of independent interest. One is the “Genericity Theorem” 6.1 which says that the essential dimension of a smooth integral Deligne–Mumford stack satisfying an appropriate separation hypothesis is the sum of its dimension and the essential dimension of its generic gerbe. This somewhat surprising result implies that the essential dimension of a non-empty open substack equals the essential dimension of the stack. In particular, it proves Theorem 1.2 in the cases where a general curve in M_{g,n} has no non-trivial automorphisms. It also brings into relief the important role played by gerbes in this theory.

The second main ingredient in our proof of Theorem 1.2 is the following formula, which we use to compute the essential dimension of the generic gerbe.

Theorem 1.3. Let X be a gerbe over a field K banded by a group G. Let [X] ∈ H^{2}(K, G) be the Brauer class of X.

(a) If G = 𝔾_{m} and ind[X] is a prime power then ed X = ind [X] − 1.

(b) If G = μ_{p^{r}}, where p is a prime and r ≥ 1, then ed X = ind [X].

Our proof of this theorem can be found in the preprint [BRV07, Section 7]. A similar argument was used by N. Karpenko and A. Merkurjev in the proof of [KM08, Theorem 3.1], which generalizes Theorem 1.3(b). For the sake of completeness, we include an alternative proof of Theorem 1.3 in §4.

Theorem 1.3 has a number of applications beyond Theorem 1.2. Some of these have already appeared in print. In particular, we used Theorem 1.3 to study the essential dimension of spinor groups in [BRV], N. Karpenko and A. Merkurjev [KM08] used it to study the essential dimension of finite p-groups, and A. Dhillon and N. Lemire [DL09] used it, in combination with the Genericity Theorem 6.1, to give an upper bound for the essential dimension of the moduli stack of SL_{n}-bundles over a projective curve. In this paper we will also use Theorem 1.3 (in combination with Theorems 6.1) to study the essential dimension of the stacks of hyperelliptic curves (Theorem 7.2) and principally polarized abelian varieties (Theorem 9.1).

In the case where (g, n) = (1, 0) Theorem 1.2 requires a separate argument, which is carried out in §8. In this case Theorem 1.2 is a consequence of the fact that the group schemes of l^{n}-torsion points on a Tate curve has essential dimension l^{n}, where l is a prime.

Acknowledgments. We would like to thank the Banff International Research Station in Banff, Alberta (BIRS) for providing the inspiring meeting place where this work was started. We are grateful to J. Alper, K. Behrend, C.-L. Chai, D. Edidin, N. Fakhruddin, A. Merkurjev, B. Noohi, G. Pappas, M. Reid and B. Totaro for helpful conversations.

2. The essential dimension of a stack

Let k be a field. We will write Fields_{k} for the category of field extensions K/k. Let F: Fields_{k} → Sets be a covariant functor.

Definition 2.1. Let a ∈ F (L), where L is an object of Fields_{k}. Suppose that K ⊂ L is a field extension of k. We say that a descends to K or that K is a field of definition for a if a is in the image of the induced map F (K) → F (L).

The essential dimension ed a of a ∈ F (L) is the minimum of the transcendence degrees tr deg_{k} K taken over all fields k ⊆ K ⊆ L such that a descends to K.

The essential dimension ed F of the functor F is the supremum of ed a taken over all a ∈ F (L) with L in Fields_{k}. We will write ed F = −∞ if F is the empty functor.

These notions are relative to the base field k. To emphasize this, we will sometimes write ed_{k} a or ed_{k} F instead of ed a or ed F, respectively.

The following definition singles out a class of functors that is sufficiently broad to include most interesting examples, yet “geometric” enough to allow one to get a handle on their essential dimension.

Definition 2.2. Suppose X is an algebraic stack over k. The essential dimension ed X of X is defined to be the essential dimension of the functor F_{X} : Fields_{k} → Sets which sends a field L/k to the set of isomorphism classes of objects in X (L).^{1}

As in Definition 2.1, we will write ed_{k} X when we need to be specific about the dependence on the base field k. Similarly for ed_{k} ξ, where ξ is an object of F_{X}.

Example 2.3. Let G be an algebraic group defined over k and X = B_{k}G be the classifying stack of G. Then F_{X} is the Galois cohomology functor sending K to the set H^{1}(K, G) of isomorphism classes of G-torsors over Spec(K), in the fppf topology. The essential dimension of this functor is a numerical invariant of G, which, roughly speaking, measures the complexity of G-torsors over fields. This number is usually denoted by ed_{k} G or (if k is fixed throughout) simply by ed G; following this convention, we will often write ed G in place of ed B_{k}G. Essential dimension was originally introduced and has since been extensively studied in this context; see e.g., [BR97, Rei00, RY00, Kor00, Led02, JLY02, BF03, Lem04, CS06, Gar]. The more general Definition 2.1 is due to A. Merkurjev; see [BF03, Proposition 1.17].

Example 2.4. Let X = X be a scheme of finite type over a field k, and let F_{X}: Fields_{k} → Sets denote the functor given by K ↦ X(K). Then an easy argument due to Merkurjev shows that ed F_{X} = dim X; see [BF03, Proposition 1.17].

In fact, this equality remains true for any algebraic space X. Indeed, an algebraic space X has a stratification by schemes X_{i}. Any K-point η: Spec K → X must land in one of the X_{i}. Thus ed X = max ed X_{i} = dim X. ♠

Example 2.5. Let X = M_{g} be the stack of smooth algebraic curves of genus g. Then the functor F_{X} sends K to the set of isomorphism classes of n-pointed smooth algebraic curves of genus g over K. Question 1.1 asks about the essential dimension of this functor in the case where n = 0.

1In the literature the functor F_{X} is sometimes denoted by X̂ or \overline{X}.

Example 2.6. Suppose a linear algebraic group G is acting on an algebraic space X over a field k. We shall write [X/G] for the quotient stack [X/G]. Recall that K-points of [X/G] are by definition diagrams of the form

(2.1)  \xymatrix{T \ar[r]^{ψ} \ar[d]^{π} & X \\ Spec(K)}

where π is a G-torsor and ψ is a G-equivariant map. The functor F_{[X/G]} associates to a field K/k the set of isomorphism classes of such diagrams.

In the case where G is a special group (recall that this means that every G-torsor over Spec(K) is split, for every field K/k) the essential dimension of F_{[X/G]} has been previously studied in connection with the so-called “functor of orbits” Orb_{X,G} given by the formula

Orb_{X,G}(K) =^{def} set of G(K)-orbits in X(K).

Indeed, if G is special, the functors F_{[X/G]} and Orb_{X,G} are isomorphic; an isomorphism between them is given by sending an object (2.1) of F_{[X/G]} to the G(K)-orbit of the point ψs: Spec(K) → X, where s: Spec(K) → T is a section of π: T → Spec(K).

Of particular interest are the natural GL_{n}-actions on 𝔸^{N} = affine space of homogeneous polynomials of degree d in n variables and on ℙ^{N−1} = projective space of degree d hypersurfaces in ℙ^{n−1}, where N = \binom{n+d−1}{d} is the number of degree d monomials in n variables. For general n and d the essential dimension of the functor of orbits in these cases is not known. Partial results can be found [BF04] and [BR05, Sections 14-15]. Additional results in this setting will be featured in a forthcoming paper.

Remark 2.7. If the functor F in Definition 2.1 is limit-preserving, a condition satisfied in all cases of interest to us, then every element a ∈ F (L) descends to a field K ⊂ L that is finitely generated over k. Thus in this case ed a is finite. In particular, ed ξ is finite for every object ξ ∈ X (K) and every field extension K/k; the limit-preserving property in this case is proved in [LMB00, Proposition 4.18],

In §3 we will show that, in fact, ed X < ∞ for a broad class of algebraic stacks X; cf. Corollary 3.4. On the other hand, there are interesting examples where ed X < ∞; see Theorem 1.2 or [BS08].

The following observation is a variant of [BF03, Proposition 1.5].

Proposition 2.8. Let X be an algebraic stack over k, and let K be a field extension of k. Then ed_{K} X_{K} ≤ ed_{k} X.

Proof. If L/K is a field extension, then the natural morphism X_{K}(L) → X (L) is an equivalence. Suppose that M/k is a field of definition for an object ξ in X (L). Let N be a composite of M and K over k. Then N is a field of definition for ξ, tr deg_{K} N ≤ tr deg_{k} M, and the proposition follows. ♠

3. A fiber dimension theorem

We now recall Definitions (3.9) and (3.10) from [LMB00]. A morphism f: X → Y of algebraic stacks (over k) is said to be representable if, for every k-morphism T → Y, where T is an affine k-scheme, the fiber product X ×_{Y} T is representable by a scheme over T. A representable morphism f: X → Y is said to be locally of finite type and of fiber dimension ≤ d if the projection X ×_{Y} T → T is also locally of finite type over T and every fiber has dimension ≤ d.

Example 3.1. Let G be an algebraic group defined over k, and let X → Y be a G-equivariant morphism of k-algebraic spaces, locally of finite type and of relative dimension ≤ d. Then the induced map of quotient stacks [X/G] → [Y/G] is representable, locally of finite type and of relative dimension ≤ d.

The following result may be viewed as a partial generalization of the fiber dimension theorem (see [Har77, Exercise II.3.22 or Proposition III.9.5]) to the setting where schemes are replaced by stacks and dimension by essential dimension.

Theorem 3.2. Let d be an integer X → Y be a representable k-morphism of algebraic stacks which is locally of finite type and of fiber dimension at most d. Let L/k be a field, ξ ∈ X(L). Then

(a) ed_{k} ξ ≤ ed_{k} f(ξ) + d, and

(b) ed_{k} X ≤ ed_{k} Y + d.

In particular, if ed_{k} Y is finite, then so is ed_{k} X.

Proof. (a) By the definition of ed_{k} f(ξ) we can find an intermediate field k ⊂ K ⊂ L and a morphism η: Spec K → Y such that tr deg_{k} K ≤ ed f(ξ) and the following diagram commutes.

\xymatrix{Spec L \ar[r]^{ξ} \ar[d] & X \ar[d]^{f} \\ Spec K \ar[r]^{η} & Y}

Let X_{K} =^{def} X ×_{Y} Spec K. By the hypothesis, X_{K} is an algebraic space, locally of finite type over K and of relative dimension at most d. By the commutativity of the diagram above, the morphism ξ: Spec L → X factors through X_{K}:

\xymatrix{Spec L \ar[drr]^{ξ} \ar[dr]^{ξ_{0}} \ar[ddr] \\  & X_{K} \ar[r] \ar[d] & X \ar[d]^{f} \\  & Spec K \ar[r]^{η} & Y}

Moreover, ξ factors through K(p), where p denotes the image of ξ_{0} in X_{K}. Since X_{K} has dimension at most d over K, we have tr deg_{K} K(p) ≤ d. Therefore,

tr deg_{k} K(p) = tr deg_{K} K(p) + tr deg_{k} K ≤ ed f(ξ) + d

and part (a) follows.

Part (b) follows from (a) by taking the maximum on both sides over all L/k and all ξ ∈ X (L). ♠

Corollary 3.3. Consider an action of an algebraic group G on an algebraic space X, defined over a field k. Assume X is locally of finite type over k. Then

ed_{k} G ≥ ed_{k}[X/G] − dim X .

Proof. The natural G-equivariant map X → Spec k gives rise to a map [X/G] → B_{k}G of quotient stacks. This latter map is locally of finite type and of relative dimension ≤ dim X; see Example 3.1. Applying Theorem 3.2(b) to this map, we obtain the desired inequality. ♠

Corollary 3.4. (Finiteness of essential dimension) Let X be an algebraic stack of finite type over k. Suppose that for any algebraically closed extension Ω of k and any object ξ of X (Ω) the group scheme \underline{Aut}_{Ω}(ξ) → Spec Ω is affine. Then ed_{k} X < ∞.

Note that Corollary 3.4 fails without the assumption that all the \underline{Aut}_{Ω}(ξ) are affine. For example, by Theorem 1.2, ed M_{1,0} = +∞.

Proof. We may assume without loss of generality that X = [X/G] is a quotient stack for some affine algebraic group G acting on an algebraic space X. Indeed, by a Theorem of Kresch [Kre99, Proposition 3.5.9] X is covered by quotient stacks [X_{i}/G_{i}] of this form and hence, ed X = max_{i} ed [X_{i}/G_{i}].

If X = [X/G] then by Corollary 3.3,

ed [X/G] ≤ ed_{k} G + dim(X) .

The desired conclusion now follows from the well-known fact that ed_{k} G < ∞ for any affine algebraic group G; see [Rei00, Theorem 3.4] or [BF03, Proposition 4.11]. ♠

4. The essential dimension of a gerbe over a field

The goal of this section is to prove Theorem 1.3 stated in the Introduction. We proceed by briefly recalling some background material on gerbes from [Mil80, p. 144] and [Gir71, IV.3.1.1], and on canonical dimension from [KM06] and [BR05].

Gerbes. Let X be a gerbe defined over a field K banded by an abelian K-group scheme G. There is a notion of equivalence of gerbes banded by G; the set of equivalence classes is in a natural bijective correspondence with the group H^{2}(K, G). The identity element of H^{2}(K, G) corresponds to the class of the neutral gerbe B_{K}G. Let K be a field and let 𝔾_{m} denote the multiplicative group scheme over K. Recall that the group H^{2}(K, 𝔾_{m}) is canonically isomorphic to the Brauer group Br K of Brauer equivalence classes of central simple algebras over K.

Canonical dimension. Let X be a smooth projective variety defined over a field K. We say that L/K is a splitting field for X if X(K) ≠ ∅. A splitting field L/K is called generic if for every splitting field L_{0}/K there exists a K-place L → L_{0}. The canonical dimension cd X of X is defined as the minimal value of tr deg_{K}(L), where L/K ranges over all generic splitting fields. Note that the function field L = K(X) is a generic splitting field of X; see [KM06, Lemma 4.1]. In particular, generic splitting fields exist and cd X is finite.

The determination functor D_{X}: Fields_{K} → Sets is defined as follows. For any field extension L/K, D_{X}(L) is the empty set, if X(L) = ∅, and a set consisting of one element if X(L) ≠ ∅. The natural map D(L_{1}) → D(L_{2}) is then uniquely determined for any K ⊂ L_{1} ⊂ L_{2}. It is shown in [KM06] that if X be a complete regular K-variety then

(4.1)  cd X = ed D_{X} .

Of particular interest to us will be the case where X is a Brauer-Severi variety over K. Let m be the index of X. If m = p^{a} is a prime power then

(4.2)  cd X = p^{a} − 1 ;

see [KM06, Example 3.10] or [BR05, Theorem 11.4].

If m = p_{1}^{a_{1}} … p_{r}^{a_{r}} is the prime decomposition of m then the class of X in Br L is the sum of classes α_{1}, …, α_{r} whose indices are p_{1}^{a_{1}}, …, p_{r}^{a_{r}}. Denote by X_{1}, …, X_{r} the Brauer–Severi varieties associated with α_{1}, …, α_{r}. It is easy to see that K(X_{1} × ⋯ × X_{r}) is a generic splitting field for X. Hence,

cd X ≤ dim(X_{1} × ⋯ × X_{r}) = p_{1}^{a_{1}} + ⋯ + p_{r}^{a_{r}} − r .

J.-L. Colliot-Thélène, N. Karpenko and A. Merkurjev [CTKM06] conjectured that equality holds, i.e.,

(4.3)  cd X = p_{1}^{a_{1}} + ⋯ + p_{r}^{a_{r}} − r .

As we mentioned above, this in known to be true if m is a prime power (i.e., r = 1). Colliot-Thélène, Karpenko and Merkurjev also proved (4.3) for m = 6; see [CTKM06, Theorem 1.3]. Their conjecture remains open for all other m.

Theorem 4.1. Let d be an integer with d > 1. Let K be a field and x ∈ H^{2}(K, μ_{d}). Denote the image of x in H^{2}(K, 𝔾_{m}) by y, the μ_{d}-gerbe associated with x by X → Spec(K), the 𝔾_{m}-gerbe associated with y by Y → Spec(K), and the Brauer–Severi variety associated with y by P. Then (a) ed Y = cd P and

(b) ed X = cd P + 1. In particular, if the index of x is a prime power p^{r} then ed Y = p^{r} − 1 and ed X = p^{r}.

Proof. The last assertion follows from (a) and (b) by (4.2).

(a) The functor F_{Y}: Fields_{K} → Sets sends a field L/K to the empty set, if P (L) = ∅, and to a set consisting of one point, if P (L) ≠ ∅. In other words, F_{Y} is the determination functor D_{P} introduced above. The essential dimension of this functor is cd P; see (4.1).

(b) First note that the natural map X → Y is of finite type and representable of relative dimension ≤ 1. By Theorem 3.2(b) we conclude that ed X ≤ ed Y + 1. By part (a) it remains to prove the opposite inequality, ed X ≥ ed Y + 1. We will do this by constructing an object α of X whose essential dimension is ≥ ed Y + 1.

We will view X as a torsor for B_{K}μ_{d} in the following sense. There exist maps

X × B_{K}μ_{d} ⟶ X
X × X ⟶ B_{K}μ_{d}

satisfying various compatibilities, where the first map is the “action” of B_{K}μ_{d} on X and the second map is the “difference” of two objects of X. For the definition and a discussion of the properties of these maps, see [Gir71, Chapter IV, Sections 2.3, 2.4 and 3.3]. (Note that, in the notation of Giraud’s book, X ∧ B_{K}μ_{d} ≅ X and the action operation above arises from the map X × B_{K}μ_{d} → X ∧ B_{K}μ_{d} given in Chapter IV, Proposition 2.4.1. The difference operation, which we will not use here, arises similarly from the fact that, in Giraud’s notation, HOM(X , X) ≅ B_{K}μ_{d}.)

Let L = K(P) be the function field of P. Since L splits P, we have a natural map a: Spec L → Y. Moreover since L is a generic splitting field for P,

(4.4)  ed a = cd P = ed Y,

where we view a as an object in Y. Non-canonically lift a: Spec L → Y to a map Spec L → X (this can be done, because X → Y is a 𝔾_{m}-torsor). Let Spec L(t) → B_{L}μ_{d} denote the map classified by (t) ∈ H^{1}(L(t), μ_{d}) = L(t)^{×}/L(t)^{×d}. Composing these two maps, we obtain an object

α: Spec L(t) → X × B_{L}μ_{d} → X .

in X (L(t)). Our goal is to prove that ed α ≥ ed Y + 1. In other words, given a diagram of the form

(4.5)  \xymatrix{Spec L(t) \ar[r]^{α} \ar[d] & X \\ Spec M \ar[ur]^{β}}

where K ⊂ M ⊂ L is an intermediate field, we want to prove the inequality tr deg_{K}(M) ≥ ed Y + 1. Assume the contrary: there is a diagram as above with tr deg_{K}(M) ≤ ed Y. Let ν: L(t)^{∗} → ℤ be the usual discrete valuation corresponding to t and consider two cases.

Case 1. Suppose the restriction ν_{|M} of ν to M is non-trivial. Let M_{0} denote the residue field of ν and M_{≥0} denote the valuation ring. Since Spec M → X → Y, there exists an M-point of P. Then by the valuative criterion of properness for P, there exists an M_{≥0}-point and thus an M_{0}- point of P. Passing to residue fields, we obtain the diagram

\xymatrix{Spec L \ar[r]^{a} \ar[d] & Y \\ Spec M_{0} \ar[ur] & .}

which shows that ed a ≤ tr deg_{K} M_{0} = tr deg_{K} M − 1 ≤ ed Y − 1, contradicting (4.4).

Case 2. Now suppose the restriction of ν to M is trivial. The map Spec L → X sets up an isomorphism X_{L} ≅ B_{L}μ_{d}. The map Spec L(t) → X factors through X_{L} and thus induces a class in B_{L}μ_{d}(L(t)) = H^{1}(L(t), μ_{d}). This class is (t). Tensoring the diagram (4.5) with L over K, we obtain

\xymatrix{Spec L(t) ⊗ L \ar[r]^{α} \ar[d] & X_{L} ≅ B_{L}μ_{d} \\ Spec M ⊗ L \ar@{-->}[ur]^{β}}

Recall that L = K(P) is the function field of P. Since P is absolutely irreducible, the tensor products L(t) ⊗ L and M ⊗ L are fields. The map Spec M ⊗ L → B_{L}μ_{d} is classified by some m ∈ (M ⊗ L)^{×}/(M ⊗ L)^{×d} = H^{1}(M ⊗ L, μ_{d}). The image of m in L(t) ⊗ L would have to be equal to t modulo d-th powers. We will now derive a contradiction by comparing the valuations of m and t.

To apply the valuation to m, we lift ν from L(t) to L(t) ⊗ L. That is, we define ν_{L} as the valuation on L(t) ⊗ L = (L ⊗ L)(t) corresponding to t. Since ν_{L}(t) = ν(t) = 1, we conclude that ν_{L}(m) ≡ 1 (mod d). This shows that ν_{L} is not trivial on M ⊗ L and thus ν is not trivial on M, contradicting our assumption. This contradiction completes the proof of part (b). ♠ Corollary 4.2. Let 1 → Z → G → Q → 1 denote an extension of group schemes over a field k with Z central and isomorphic to (a) 𝔾_{m} or (b) μ_{p^{r}} for some prime p and some r > 1. Let ind(G, Z) as the maximal value of ind(∂_{K}(t)) as K ranges over all field extensions of k and t ranges over all torsors in H^{1}(K, Q). If ind(G, Z) is a prime power (which is automatic in case (b)) then

ed_{k} B_{k}G ≥ ind(G, Z) − dim G.

Proof. Choose t ∈ H^{1}(K, Z) so that ind(∂_{K}(t)) attains its maximal value, ind(G, Z). Let X → Spec(K) be the Q-torsor representing t. Then G acts on X via the projection G → Q, and [X/G] is the Z-gerbe over Spec(K) corresponding to the class ∂_{K}(t) ∈ H^{2}(K, Z). By Theorem 1.3.

ed[X/G] = \begin{cases}ind(∂_{K}(t)) − 1 in case (a), \\ ind(∂_{K}(t)) in case (b). \end{cases}

Since dim(X) = dim(Q), applying Corollary 3.3 to the G-action on X, we obtain

ed_{K} G_{K} ≥ \begin{cases}(ind(G, Z) − 1) − dim Q = ind(G, Z) − dim(G) in case (a), \\ ind(G, Z) − dim Q = ind(G, Z) − dim(G) in case (b). \end{cases}

Since ed_{k} G ≥ ed_{K} G_{K} (see [BF03, Proposition 1.5] or our Proposition 2.8), the corollary follows. ♠

5. Gerbes over complete discrete valuation rings

In this section we prove two results on the structure of étale gerbes over complete discrete valuation rings that will be used in the proof of Theorem 6.1.

5.1. Big and small étale sites. Let S be a scheme. We let Sch/S denote the category of all schemes T equipped with a morphism to S. As in [SGA72], we equip Sch/S with the étale topology. Let ét/S denote the full subcategory of Sch/S consisting of all schemes étale over S (also equipped with the étale topology). The site Sch/S is the big étale site and the category ét/S is the small étale site. We let S_{Ét} denote the category of sheaves on Sch/S and S_{ét} the category of sheaves on ét/S. Since the obvious inclusion functor from the small to the big étale site is continuous, it induces a continuous morphism of sites u: ét/S → Sch/S and, thus, a morphism u: S_{Ét} → S_{ét}. Moreover, the adjunction morphism F → u_{∗}u^{∗}F is an isomorphism for F a sheaf in S_{ét} [SGA72, VII.4.1]. We can therefore regard S_{ét} as a full subcategory of S_{Ét}.

Definition 5.1. Let S be a scheme. An étale gerbe over S is a separated locally finitely presented Deligne–Mumford stack over S that is a gerbe in the étale topology.

Let X → S be an étale gerbe over a scheme S. Then, by definition, there is an étale atlas, i.e., a morphism U_{0} → X where U_{0} → S is surjective, étale and finitely presented over S. This atlas gives rise to a groupoid G =^{def} [U_{1} =^{def} U_{0} ×_{X} U_{0} ⇉ U_{0}] in which each term is étale over S. Since X is the stackification of G which is a groupoid on the small étale site S_{ét}, it follows that X = u^{∗}X′ for X′ a gerbe on S_{ét}. In other words, we have the following proposition.

Proposition 5.2. Let X → S be an étale gerbe over a scheme S. Then there is a gerbe X′ on S_{ét} such that X = u^{∗}X′.

In fact, when S is the spectrum of a henselian discrete valuation ring, we can do better by using the following

Proposition 5.3. Let S = Spec R with R a henselian discrete valuation ring. Let f: T → S be a surjective étale morphism. Then there is an open component T′ of T such that f_{|T′}: T′ → S is a finite étale morphism

Proof. Let s denote the closed point of S. Since f is surjective, there exists a t ∈ T such that f(t) = s. Since f is étale, f is quasi-finite at t (by [Gro67, 17.6.1]. Now, it follows from [Gro67, 18.5.11] that T′ =^{def} Spec O_{T,t} is an open component of T which is finite and étale. ♠

Now for a scheme S, let fét/S denote the category of finite étale covers T → S. We can consider fét/S as a site in the obvious way. Then the inclusion morphism induces a continuous morphism of sites v: ét/S → fét/S. If S is a henselian trait (i.e., the spectrum of a henselian discrete valuation ring) with closed point s, then the inclusion morphism i: s → S induces an equivalence of categories i^{∗}: fét/S → fét/s. Since the site fét/s is equivalent to s_{ét}, this induces the specialization morphism sp: S_{ét} → s_{ét}, which is inverse to the inclusion morphism i: s_{ét} → S; cf. [SGA73, p. 89]. Let τ = sp ∘u: S_{Ét} → s_{ét}.

Corollary 5.4. Let X → S be an étale gerbe over a henselian trait S with closed point s. Then there is a gerbe X′′ over s_{ét} such that X = τ^{∗}X′′.

Proof. Since X → S is an étale gerbe, there is an étale atlas X_{0} → S of X. By Proposition 5.3 we may assume that X_{0} such that X′_{0} is finite over S; and then X_{1} =^{def} X_{0} ×_{X} X_{0} is also finite, because X is separated, by hypothesis. Now the equivalence of categories i^{∗}: fét/S → fét/s produces an gerbe X′′ over s_{ét} such that X = τ^{∗}X′′. ♠

5.2. Group extensions and gerbes. Let k be a field with separable closure \overline{k} and absolute Galois group G = Gal(\overline{k}/k). Let

(5.1)  1 ⟶ F ⟶^{i} E ⟶^{p} G ⟶ 1

be an extension of profinite groups with F finite and all maps continuous. From this data, we can construct a gerbe X_{E} over (Spec k)_{ét}. To determine the gerbe it is enough to give its category of sections over Spec L where L/k is a finite separable extension. Let K = {g ∈ G | g(α) = α, α ∈ L}. Then the objects of the category X_{E}(L) are the solutions of the embedding problem given by (5.1). That is, an object of X_{E}(L) is a continuous homomorphism σ: K → E such that p ∘ σ(k) = k for k ∈ K. If s_{i}: K → E, i = 1, 2 are two objects in X_{E}(L) then a morphism from s_{1} to s_{2} is an element f ∈ F such that fs_{1}f^{−1} = s_{2}; cf. [DD99, p. 581].

By the results of Giraud [Gir71, Chapter VIII], it is easy to see that any gerbe X → Spec k with finite inertia arises from a sequence (5.1) as above. We explain how to get the extension: Given X, we can find a separable Galois extension L/k and an object ξ ∈ X (L). This gives an extension of groups Aut_{X} (ξ) → Aut_{Spec k}(Spec L) = Gal(L/k). Pulling back this extension via the map G = Gal(k) → Gal(L/k) gives the desired extension (5.1).

Now, suppose that E is an extension of groups as in (5.1). Let L/k be a field extension, which is separable but not necessarily finite. Let \overline{L} denote a fixed separable closure of L and let \overline{k} denote the separable closure of k in \overline{L}. Then there is an obvious map r: Gal(\overline{L}/L) → Gal(\overline{k}/k). Let u: (Spec k)_{Ét} → (Spec k)_{ét} denote the functor of §5.1. Then u^{∗}X (L) has the same description as in the case where L is a finite extension of k. In other words, we have the following proposition.

Proposition 5.5. Let L/k be a separable extension and let X_{E} be the gerbe as above. Then the objects of the category u^{∗}X_{E}(L) are the morphisms s: Gal(L) → Gal(k) making the following diagram commute.

\xymatrix{ &  &  & Gal(L) \ar[dl]_{s} \ar[d]^{r} \\ 1 \ar[r] & F \ar[r] & E \ar[r] & Gal(k) \ar[r] & 1}

Moreover, if s_{i}, i = 1, 2 are two objects in u^{∗}X_{E}(L), then the morphisms from s_{1} to s_{2} are the elements f ∈ F such that fs_{1}f^{−1} = s_{2}.

5.3. Splitting the inertia sequence. We begin by recalling some results and notation from Serre’s chapter in [GMS03].

Let A be a discrete valuation ring. Write S = S_{A} for Spec A, s = s_{A} for the closed point in S and η = η_{A} for the generic point. When A is the only discrete valuation ring under consideration, we suppress the subscripts. If A is henselian, then the choice of a separable closure k(\overline{η}) of k(η) induces a separable closure of k(s) and a map Gal(k(\overline{η})) → Gal(k(\overline{s})) between the absolute Galois groups. The kernel of this map is called the inertia, written as I = I_{A}. If char k(s) = p > 0, then we set I^{w} = I_{A}^{w} equal to the unique p-Sylow subgroup of I; otherwise we set I^{w} = {1}. The group I^{w} is called the wild inertia. The group I_{t} = I_{A,t} =^{def} I/I^{w} is called the tame inertia and the group Gal(k(η))_{t} =^{def} Gal(k(η))/I^{w} is called the tame Galois group. We therefore have sequences

(5.2) 1 ⟶ I ⟶ Gal(k(η)) ⟶ Gal(k(s)) ⟶ 1 and (5.3) 1 ⟶ I_{t} ⟶ Gal(k(η))_{t} ⟶ Gal(k(s)) ⟶ 1 ,

called the inertia exact sequence and the tame inertia exact sequence, respectively.

For each prime l, set ℤ_{l}(1) = \underleftarrow{lim} μ_{l^{n}} so that

∏_{l≠p} ℤ_{l}(1) = \underleftarrow{lim}_{p ∤ n}μ_{n}.

Then there is a canonical isomorphism c: I_{t} → ∏_{l≠p} ℤ_{l}(1) [GMS03, p. 17]. To explain this isomorphism, let g ∈ I_{t} and let π^{1/n} be an n-th root of a uniformizing parameter π ∈ A with n not divisible by p. Then the image of c(g) in μ_{n} is g(π^{1/n})/π^{1/n}.

The following result extends Lemma 7.6 in [GMS03] from the case of complete to henselian discrete valuation rings.

Proposition 5.6. Let A be a henselian discrete valuation ring. Then the sequence (5.2) is split.

Proof. Because we need the ideas from the proof, we will repeat Serre’s argument. Set K = k(η) and \overline{K} = k(\overline{η}). Set K_{t} = \overline{K}^{I_{t}}: the maximal tamely ramified extension of K. Let π be a uniformizing parameter in A. Then, for each non-negative integer n not divisible by p, choose an n-th root π_{n} of π in \overline{K} such that π_{nm}^{m} = π_{n}. Set K_{ram} =^{def} K[π_{n}]_{(p ∤ n)}. Then K_{ram} is totally and tamely ramified over K. Moreover any K_{t} = K_{ram}K_{unr}. It follows that Gal(k(s)) map be identified with the subgroup of elements g ∈ Gal(K)_{t} fixing each of the π_{n}; cf. [Del80]. This splits the sequence (5.3).

Now, in [GMS03], Serre extends this splitting non-canonically to a splitting of (5.2) as follows. Since k(s) has characteristic p, the p-cohomological dimension of Gal(k(s)) is ≤ 1; see [Ser02]. Consequently, any homomorphism Gal(k(s)) → Gal(K)_{t} can be lifted to Gal(K). ♠

While the splitting of (5.3) is not canonical, we need to know that it is possible to split two such sequences, associated to henselian discrete valuation rings A ⊆ B, in a compatible way.

Proposition 5.7. Let A ⊆ B be an extension of henselian discrete valuation rings, such that a uniformizing parameter for A is also a uniformizing parameter for B. Then it is possible to find maps σ_{B}: Gal(k(s_{B})) → Gal(k(η_{B}))_{t} (resp. σ_{A}: Gal(k(s_{A})) → Gal(k(η_{A}))_{t}) splitting the tame inertia exact sequence (5.3) for B (resp. A) and such that the diagram

\xymatrix{Gal(k(s_{B})) \ar[r]^{σ_{B}} \ar[d] & Gal(k(η_{B}))_{t} \ar[d] \\ Gal(k(s_{A})) \ar[r]^{σ_{A}} & Gal(k(η_{A}))_{t},}

with vertical morphisms given by restriction, commutes.

Proof. Let π ∈ A be a uniformizing parameter for A, and hence for B. For each n not divisible by p = char(k(s_{A})), choose an n-th root π_{n} of π in k(\overline{η}_{B}). Now, set σ_{B}(k(s_{B})) = \{g ∈ Gal(k(η_{B}))_{t} : g(π_{n}) = π_{n} for all n\} and similarly for A. By the proof of Proposition 5.6, this defines splitting of the tame inertia sequences. Moreover, these splittings lift to splittings of the inertia exact sequence. ♠

Remark 5.8. By the proof of Proposition 5.6, the splittings σ_{B} and σ_{A} in Proposition 5.7 can be lifted to maps σ̃_{B}: Gal(k(s_{B})) → Gal(k(η_{B})) (resp. σ̃_{A}: Gal(k(s_{A})) → Gal(k(η_{a}))). However, since these liftings are non-canonical it is not clear that σ̃_{B} and σ̃_{A} can be chosen compatibly.

5.4. Tame gerbes and splittings. The following result is certainly well known; for the sake of completeness we supply a short proof.

Proposition 5.9. Let X → S be an étale gerbe over a henselian trait, with closed point s. Let i: s → S denote the inclusion of the closed point and sp: S → s denote the specialization map. Then restriction map

i^{∗}: X (S) ⟶ X (s)

induces an equivalence of categories with quasi-inverse given by

sp^{∗}: X (s) ⟶ X (S) .

Proof. Since the composite s → S →^{sp} s is an auto-equivalence and X is obtained by pullback from X_{s}, it suffices to show that the functor i^{∗}: X (S) → X (s) is faithful. For this, suppose ξ_{i}: S → X , i = 1, 2 are two objects of X (S). Then the sheaf Hom(ξ_{1}, ξ_{2}) is étale over S. Since S is henselian, it follows that the sections of Hom(ξ_{1}, ξ_{2}) over S are isomorphic (via restriction) to the sections over s. Thus i^{∗}: X (S) → X (s) is fully faithful. ♠

A Deligne-Mumford stack X → S is tame if, for every geometric point ξ: Spec Ω → X, the order of the automorphism group Aut_{Spec Ω}(ξ) has order prime to the characteristic of Ω. For tame gerbes over a henselian discrete valuation ring, we have the following analogue of the splitting in Proposition 5.7.

Theorem 5.10. Let A ⊆ B be an extension of henselian traits, such that a uniformizing parameter for A is sent to a uniformizing parameter for B, and let X be a tame étale gerbe over A. Write j_{B}: {η_{B}} → B (resp. j_{A}: {η_{A}} → A) for the inclusion of the generic points. Then it is possible to find functors σ_{B}: X (k(η_{B})) → X (B) and σ_{A}: X (k(η_{A})) → X (A) such that the diagram

\xymatrix{X (A) \ar[r]^{j_{A}^{∗}} \ar[d]^{h^{∗}} & X (k(η_{A})) \ar[r]^{σ_{A}} \ar[d]^{h^{∗}} & X (A) \ar[d]^{h^{∗}} \\ X (B) \ar[r]^{j_{B}^{∗}} & X (k(η_{B})) \ar[r]^{σ_{B}} & X (B) ,}

where h: Spec B → Spec A is the morphism induced by the inclusion A ⊆ B, commutes (up to natural isomorphism) and the horizontal composites are isomorphic to the identity.

Proof. Since X is an étale gerbe, there is an extension E as in (5.1) with G = Gal(k(s_{A})) such that X is the pull-back of X_{E} to the big étale site over S_{A}. Since X is tame, the band, i.e., the group F in (5.1), has order prime to char k(s_{A}).

Now, pick splittings σ_{B} and σ_{A} compatibly, as in Proposition 5.7.

We define a functor X (k(η_{A})) → X (k(η_{B})) as follows. Using Proposition 5.5 we can identify X (k(η_{B})) with category of sections s: Gal(kη_{B})) → E. Given such a section s, the tameness of E implies that s(I^{w}) = 1. Therefore, s induces a map Gal(k(η_{B})_{t} → Gal(k(s_{B}) (which we also denote by the symbol s). Let σ_{B}(s) denote the section s ∘ σ_{B}: Gal(k(s_{B})) → X_{B}. This defines σ_{B} on the objects in X (k(η_{B})). If we define σ_{A} in the same way, it is clear that the diagram above commutes on objects. We define σ_{B} (resp. σ_{A}) on morphisms, by setting σ_{B}(f) = f (and similarly for A). We leave the rest of the verification to the reader. ♠

5.5. Genericity.

Theorem 5.11. Let R be a discrete valuation ring, S = Spec(R) and

X ⟶ S

a tame étale gerbe. Then ed_{k(s)} X_{s} ≤ ed_{k(η)} X_{η}, where s is the closed point of S and η is the generic point.

Proof. First note that we can assume that R is complete. If not, replace R with its completion at s. The field k(s) does not change, but k(η) is replaced by a field extension. By Proposition 2.8, the essential dimension of X_{k(η)} does not increase.

If R is equicharacteristic, then by Cohen’s structure theorem, R = k[[t]] with k = k(s). If not, denote by W (k(s)) the unique complete discrete valuation ring with residue field k(s) and uniformizing parameter p. This is called a Cohen ring of k(s) in [Gro64, 19.8]. If k(s) is perfect then W (k(s)) is the ring of Witt vectors of k(s), but this is not true in general, and W (k(s)) is only determined up to a non-canonical isomorphism. By [Gro64, Théorème 19.8.6], there is a homomorphism W (k(s)) → R inducing the identity on k(s); since X is pulled back from k(s) via the specialization map, we can replace R with W (k(s)).

Now suppose b: Spec L → X_{s} is a morphism from the spectrum of a field with ed_{k(s)} b = tr deg_{k(s)} L = ed X_{s}. (There is such a morphism because the ed X_{s} is finite.) Set B = L[[t]] in the case that R is equicharacteristic and equal to W (L) otherwise. In either case, B is a complete discrete valuation ring with residue field L. In the first case we have a canonical embedding R = k[[t]] ⊆ L[[t]] = B; in the second case, again by [Gro64, Théorème 19.8.6] (due to Cohen), we have a lifting R = W (k(s)) → W (L) = B of the embedding k(s) ⊆ L, which is easily seen to be injective. Therefore there is a unique morphism β = b ∘ sp: S_{B} → X whose specialization to the closed point of B coincides with ξ.

Suppose there is a subfield M of k(η_{B}) containing k(η_{R}) such that the following hold

(1) The restriction j_{B}^{∗}β of β to k(η_{B}) factors through M;

(2) tr deg_{k(η_{R})} M < ed_{k(s)} b. Complete M with respect to the discrete valuation induced from k(η_{B}) and call the resulting complete discrete valuation ring A. It follows that there is a class α in X (k(η_{A})) whose restriction to k(η_{B}) coincides with j_{B}^{∗}β. But then, by Theorem 5.10, we have β = h_{∗}σ_{A}(α). This implies that b: Spec L → X_{s} factors through the special fiber of A. Since the transcendence degree of k(s_{A}) over k(s) is less than ed_{k(s)} b, this is a contradiction. ♠

Corollary 5.12. Let R be an equicharacteristic complete discrete local ring and X → Spec(R) be a tame étale gerbe. Then

ed_{k(s)} X_{s} = ed_{k(η)} X_{η} ,

where s denotes the closed point of Spec(R) and η denotes the generic point.

Proof. Set k = k(s). Since R is equicharacteristic, we have R = k[[t]] and X_{k(η)} is the pullback to k(η) of X_{k(s)} via the inclusion of k in k((t)). Therefore ed_{k(s)} X_{k(s)} ≥ ed_{k(η)} X_{k(η)}. The opposite inequality is given by Theorem 5.11.

♠

Theorem 5.13. Suppose that X is an étale gerbe over a smooth scheme X locally of finite type over a field k of characteristic 0. Let K be an extension of k, ξ ∈ X (Spec K). Then

ed ξ ≤ ed_{k(X)} X_{k(X)} + dim X − codim_{X} ξ.

Proof. We proceed by induction on codim ξ. If codim_{X} ξ = 0, then the morphism ξ: Spec K → X is dominant. Hence ξ factors through X_{k(X)}, and the result is obvious.

Assume codim_{X} ξ > 0. Call Y the closure of the image of Spec K in X; then Y is generically smooth over Spec k, because k is perfect. By restricting to a neighborhood of the generic point of Y, we may assume that Y is contained in a smooth hypersurface X′ of X. Denote by Y and X′ the inverse images in X of Y and X′ respectively. Set R = O_{X,Y} and call X the pullback of X to R. Then we can apply Theorem 5.11 to the gerbe X_{R} → Spec R and conclude that

ed_{k(X′)} X′_{k(X′)} ≤ ed_{k(X)} X_{k(X)}.

Using the inductive hypothesis we have

ed ξ ≤ ed_{k(X′)} X_{k(X′)} + dim X′ − codim_{X′} ξ
≤ ed_{k(X)} X_{k(X)} + dim X − 1 − codim_{X′} ξ
≤ ed_{k(X)} X_{k(X)} + dim X − codim_{X} ξ.  ♠

6. A genericity theorem for a smooth Deligne–Mumford stack

It is easy to see that Theorem 5.13 fails if X is not assumed to be a gerbe. In this section we will use Theorem 5.13 to prove the following weaker result for a wider class of Deligne–Mumford stacks.

Recall that a Deligne–Mumford stack X over a field k is tame if the order of the automorphism group of any object of X over an algebraically closed field is prime to the characteristic of k.

Theorem 6.1. Let X be a smooth integral tame Deligne–Mumford stack locally of finite type over a perfect field k. Then

ed X = ed_{k(X)} X_{k(X)} + dim X .

Here the dimension of X is the dimension of the moduli space of any non-empty open substack of X with finite inertia.

Before proceeding with the proof, we record two immediate corollaries.

Corollary 6.2. If X is as above and U is an open dense substack, then ed_{k} M = ed_{k} U. ♠

Corollary 6.3. If the conditions of the Theorem 6.1 are satisfied, and the generic object of X has no non-trivial automorphisms (i.e., X is an orbifold, in the topologists’ terminology), then ed_{k} X = dim X.

Proof. Here the generic gerbe X_{K} is a scheme, so ed_{K} X_{K} = dim X. ♠

Proof of Theorem 6.1. The inequality ed X ≥ ed_{k(X)} X_{k(X)} + dim X is obvious: so we only need to prove the inequality

(6.1)  ed ξ ≤ ed_{k(X)} X_{k(X)} + dim X

for any field extension L of k and any object ξ of X (L).

First of all, let us reduce the general result to the case that X has finite inertia. The reduction is immediate from the following lemma, that is essentially due to Keel and Mori.

Lemma 6.4 (Keel–Mori). There exists an integral Deligne–Mumford stack with finite inertia X′, together with an étale representable morphism of finite type X′ → X, and a factorization Spec L → X′ → X of the morphism Spec L → X corresponding to ξ.

Proof. We follow an argument due to Conrad. By [Con, Lemma 2.2] there exist

(i) an étale representable morphism W → X such that every morphism Spec L → X, where L is a field, lifts to Spec L → W, and

(ii) a finite flat representable map Z → W, where Z is a scheme.

Condition (ii) implies that W is a quotient of Z by a finite flat equivalence relation Z ×_{W} Z ⇉ Z, which in particular tells us that W has finite inertia. We can now take X′ to be a connected component of W containing a lifting Spec L → W of Spec L → X. ♠

Suppose that we have proved the inequality (6.1) whenever X has finite inertia. If denote by ξ′ the object of X′ corresponding to a lifting Spec L → X′, we have

ed ξ ≤ ed ξ′ ≤ ed_{k(X′)} X′_{k(X′)}.

On the other hand, the morphism X′_{k(X′)} → X_{k(X)} induced by the étale representable morphism X′ → X is representable with fibers of dimension 0, hence

ed_{k(X′)} X′_{k(X′)} = ed_{k(X)} X′_{k(X′)} ≤ ed_{k(X)} X_{k(X)}

by Theorem 3.2 (the first equality follows immediately from the fact that the extension k(X) ⊆ k(X′) is finite).

So, in order to prove the inequality (6.1) we may assume that X has finite inertia. Denote by Y ⊆ X the closure of the image of the composite Spec L → X → X, where Spec L → X corresponds to ξ, and call Y the reduced inverse image of Y in X. Since k is perfect, Y is generically smooth; by restricting to a neighborhood of the generic point of Y we may assume that Y is smooth.

Denote by N → Y the normal bundle of Y in X. Consider the deformation to the normal bundle ϕ: M → ℙ_{k}^{1} for the embedding Y ⊆ X. This is a smooth morphism such that ϕ^{−1}𝔸_{k}^{1} = X ×_{Spec k} 𝔸_{k}^{1} and ϕ^{−1}(∞) = N, obtained as an open substack of the blow-up of X ×_{Spec k} ℙ_{k}^{1} along Y × {∞} (the well-known construction, explained for example in [Ful98, Chapter 5], generalizes immediately to algebraic stacks). Denote by M^{0} the open substack whose geometric points are the geometric points of M with stabilizer of minimal order (this is well defined because M has finite inertia).

We claim that M^{0} ∩ N ≠ ∅. This would be evident if X where a quotient stack [V/G], where G is a finite group of order not divisible by the characteristic of k, acting linearly on a vector space V, and Y where the of the form [X/G], where W is a G-invariant linear subspace of V. However, étale locally on X every tame Deligne–Mumford stack if a quotient [X/G], where G is a a finite group of order not divisible by the characteristic of k (see, e.g., [AV02, Lemma 2.2.3]). Since G is tame and X is smooth, it is well-known that étale-locally on X, the stack X has the desired form, and this is enough to prove the claim.

Set N^{0} =^{def} M^{0} ∩ N. The object ξ corresponds to a dominant morphism Spec L → Y. The pullback N ×_{Y} Spec K is a vector bundle V over Spec L, and the inverse image N^{0} ×_{Y} Spec L of N^{0} is not empty. We may assume that L is infinite; otherwise ed ξ = 0 and there is nothing to prove. Assuming that L is infinite, N^{0} ×_{Y} Spec L has an L-rational point, so there is a lifting Spec L → N^{0} of Spec L → Y, corresponding to an object η of N^{0}(Spec L). Clearly the essential dimension of ξ as an object of X is the same as its essential dimension as an object of Y, and ed ξ ≤ ed η. Let us apply Theorem 5.13 to the gerbe M^{0}. The function field of the moduli space M of M is k(X)(t), and its generic gerbe is X_{k(X)(t)}; by Proposition 2.8, we have ed_{k(X)(t)} X_{k(X)(t)} ≤ ed_{k(X)} X_{k(X)}. The composite Spec L → N^{0} ⊆ M^{0} has codimension at least 1, hence we obtain

ed ξ < ed_{k(X)(t)} X_{k(X)(t)} + dim M
≤ ed_{k(X)} X_{k(X)} + dim X + 1.

This concludes the proof. ♠

The following examples show that Theorem 6.1 fails for more general algebraic stacks, including singular Deligne–Mumford stacks.

Examples 6.5.

(a) Let k be any field. Suppose G =^{def} 𝔾_{a} × 𝔾_{a} acts on 𝔸^{3} by the formula

(s, t)(x, y, z) = (x + sz, y + tz, z), and define X =^{def} [𝔸^{3}/G]. Let H ⊆ 𝔸^{3}

be the hyperplane defined by the equation z = 0. Then X is the union of the open substack [(𝔸^{3} ∖ H)/G] ≃ 𝔸^{1} ∖ {0} and the closed substack [H/G] ≃ 𝔸^{2} × B_{k}G; hence its essential dimension is 2, its generic gerbe is trivial, and its dimension is 1.

(b) Let r and n be integers, r > 1. Assume that the characteristic of k is prime to r. Let X ⊆ 𝔸^{n} be the hypersurface defined by the equation x_{1}^{r} + ⋯ + x_{n}^{r} = 0. Let G =^{def} μ_{r}^{n} act on X via the formula

(s_{1}, …, s_{n})(x_{1}, …, x_{n}) = (s_{1}x_{1}, …, s_{n}x_{n}).

Set X = [X/G]. Then X is the union of [(X ∖ {0})/G], which is a quasi-projective scheme of dimension n − 1, and [{0}/G] ≃ B_{k}μ_{r}^{n}, which has essential dimension n.

(c) The following example shows that Corollary 6.2 fails even for quotient stacks of the form [X/G], where X is a complex affine variety and G is a connected complex reductive linear algebraic group.

Consider the action of G = GL_{n} on the affine space X of all n × nmatrices by multiplication on the left. Since G has a dense orbit, and the stabilizer of a non-singular matrix in X is trivial, we have

ed(generic gerbe of [X/G]) = 0.

On the other hand, let Y be the locus of matrices of rank n−1, which is a locally closed subscheme of X. There is a surjective GL_{n}-equivariant morphism Y → ℙ^{n−1}, sending a matrix A into its kernel, which induces a morphism [Y/G] → ℙ^{n−1}. If L is an extension of ℂ, every L-valued point of ℙ^{n−1} lifts to an L-valued point of Y: hence we have

ed [X/G] ≥ ed [Y/G] ≥ n − 1.

It is not hard to see that the essential dimension of [X/G] is the maximum of all the dimensions of Grassmannians of r planes in ℂ^{n}, which is n^{2}/4 if n is even, and (n^{2} − 1)/4 if n is odd.

Question 6.6. Under what hypotheses the genericity theorem holds? Let X → Spec k be an integral algebraic stack. Using the results of [LMB00, Chapter 11], one can define the generic gerbe X_{K} → Spec K of X, which is an fppf gerbe over a field of finite transcendence degree over k. What conditions on X ensure the equality

ed_{k} X = ed_{K} X_{K} + tr deg_{k} K?

Smoothness seems necessary; as we have seen above, there are counterexamples even for Deligne–Mumford stacks with very mild singularities. We think that the best result that one hope for is the following. Suppose that X is smooth with quasi-affine diagonal, and let ξ ∈ X (Spec L) be a point. Assume that the automorphism group scheme of ξ over L is linearly reductive. Then ed ξ ≤ ed_{K} X_{K} + tr deg_{k} K. In particular, if all the automorphism groups are linearly reductive, then ed X = ed_{K} X_{K} + tr deg_{k} K.

7. The essential dimension of M_{g,n} for (g, n) ≠ (1, 0)

Recall that the base field k is assumed to be of characteristic 0.

The assertion that ed \overline{M}_{g,n} = ed M_{g,n} whenever 2g − 2 + n > 0 is an immediate consequence of Corollary 6.2. Moreover, if g ≥ 3, or g = 2 and n ≥ 1, or g = 1 and n ≥ 2, then

ed_{k} M_{g,n} = ed_{k} \overline{M}_{g,n} = 3g − 3 + n.

Indeed, in all these cases the automorphism group of a generic object of M_{g,n} is trivial, so the generic gerbe is trivial, and ed M_{g,n} = dim M_{g,n} by Corollary 6.3.

The remaining cases of Theorem 1.2, with the exception of (g, n) = (1, 0), are covered by the following proposition. The case where (g, n) = (1, 0) requires a separate argument which will be carried out in the next section.

Proposition 7.1. (a) ed M_{0,1} = 2, (b) ed M_{0,1} = ed M_{0,2} = 0,

(c) ed M_{1,1} = 2,

(d) ed M_{2,0} = 5.

Proof. (a) Since M_{0,0} ≃ B_{k}PGL_{2}, we have ed M_{0,0} = ed PGL_{2} = 2, where the last inequality is proved in [Rei00, Lemma 9.4 (c)] (the argument there is valid for any field k of characteristic ≠ 2).

Alternative proof of (a): The inequality ed M_{0,0} ≤ 2 holds because every smooth curve of genus 0 over a field K is a conic in ℙ_{K}^{2}, and can be defined by an equation of the type ax^{2} + by^{2} + x^{2} = 0 for some a, b ∈ K, hence is defined over k(a, b). The opposite inequality follows from Tsen’s theorem.

(b) A smooth curve C of genus 0 with one or two rational points over an extension K of k is isomorphic to (ℙ_{k}^{1}, 0) or (ℙ_{k}^{1}, 0, ∞). Hence, it is defined over k.

Alternative proof of (b): M_{0,2} = B_{k}𝔾_{m} and M_{0,1} = B_{k}(𝔾_{m} ⋉ 𝔾_{a}), and the groups 𝔾_{m} and 𝔾_{m} ⋉𝔾_{a} are special (and hence have essential dimension

(c) Let M_{1,1} → 𝔸_{k}^{1} denote the map given by the j-invariant and let X denote the pull-back of M_{1,1} to the generic point Spec k(j) of 𝔸^{1}. Then X is banded by μ_{2} and is neutral by [Sil86, Proposition 1.4 (c)], and so ed_{k} X = ed_{k(j)} X + 1 = ed B_{k(j)}μ_{2} + 1 = 2.

(d) is a special case of Theorem 7.2 below, since H_{2} = M_{2,0}. ♠

Theorem 7.2. Let H_{g} denote the stack of hyperelliptic curves of genus g > 1 over a field k of characteristic 0. Then

ed H_{g} = \begin{cases}2g & if g is odd, \\ 2g + 1 & if g is even. \end{cases}

Proof. Denote by H_{g} the moduli space of H_{g}; the dimension of H_{g} is 2g − 1. Let K be the field of rational functions on H_{g}, and denote by (H_{g})_{K} =^{def} Spec K ×_{H_{g}} H_{g} the generic gerbe of H_{g}. From Theorem 6.1 we have

ed H_{g} = 2g − 1 + ed_{K}(H_{g})_{K},

so we need to show that ed_{K}(H_{g})_{K} is 1 if g is odd, 2 if g is even.

For this we need some standard facts about stacks of hyperelliptic curves, which we recall.

Let D_{g} be the stack over k whose objects over a k-scheme S are pairs (P → S, Δ), where P → S is a conic bundle (that is, a Brauer–Severi scheme of relative dimension 1), and Δ ⊆ P is a Cartier divisor that is étale of degree 2g+2 over S. Every family π: C → S in H(S) comes with a unique flat morphism C → P of degree 2, where P → S is a smooth conic bundle; denote by Δ ⊆ P its ramification locus. Sending π: C → S to (P → S, Δ) gives a morphism H_{g} → D_{g}. Recall the usual description of ramified double covers: if we split π_{∗}O_{C} as O_{P} ⊕ L, where L is the part of trace 0, then multiplication yields an isomorphism L^{⊗2} ≃ O_{P} (−Δ). Conversely, given an object (P → S, Δ) of D_{g}(S) and a line bundle L on P, with an isomorphism L^{⊗2} ≃ O_{P} (−Δ), the direct sum O_{P} ⊕ L has an algebra structure, whose relative spectrum is a smooth curve C → S with a flat map C → P of degree 2.

The morphism H_{g} → H_{g} factors through D_{g}, and the morphism D_{g} → H_{g} is an isomorphism over the non-empty locus of divisors on a curve of genus 0 with no non-trivial automorphisms (this is non-empty because g ≥ 2, hence 2g + 2 ≥ 5). Denote by (P → Spec K, Δ) the object of D_{g}(Spec K) corresponding the generic point Spec K → H_{g}. It is well-known that P (K) = ∅; we give a proof for lack of a suitable reference.

Let C be a conic without rational points defined over some extension L of k. Let V be the L-vector space H^{0}(C, ω_{C/L}^{−(g+1)}); denote the function field of

V by F = L(V). Then there is a tautological section σ of H^{0}(C_{F} , ω_{C_{F} /F}^{−(g+1)}) =

H^{0}(C, ω_{C/L}^{−(g+1)}) ⊗_{L} F. Note that C_{F} (F) = ∅, because the extension L ⊆ F is purely transcendental. The zero scheme of σ is a divisor on C_{F} that is étale over Spec F, and defines a morphism C_{F} → D_{g}. This morphism is clearly dominant: so K ⊆ F, and C_{F} = P ×_{Spec L} Spec F. Since C_{F} (F) = ∅ we have P (K) = ∅, as claimed.

By the description above, the gerbe (H_{g})_{K} is the stack of square roots of O_{P} (−Δ), which is banded by μ_{2}. When g is odd then there exists a line bundle of degree g + 1 on P, whose square is isomorphic to O_{P} (−Δ); this gives a section of (H_{g})_{K}, which is therefore isomorphic to B_{K}μ_{2}, whose essential dimension over K is 1. If g is even then such a section does not exist, and the stack is isomorphic to the stack of square roots of ω_{P/K}, whose class in H^{2}(K, μ_{2}) represents the image in H^{2}(K, μ_{2}) of the class [P] in H^{1}(K, PGL_{2}) under the non-abelian boundary map H^{1}(K, PGL_{2}) → H^{2}(K, μ_{2}). According to Theorem 1.3 its essential dimension is the index of [P], which equals 2. ♠

The results above apply to more than stable curves. Assume that we are in the stable range 2g − 2 + n > 0. Denote by 𝔐_{g,n} the stack of all reduced n-pointed local complete intersection curves of genus g. This is the algebraic stack over Spec k whose objects over a k-scheme T are finitely presented proper flat morphisms C → T, where C is an algebraic space, whose geometric fibers are connected reduced local complete intersection curves of genus g, together with n sections T → C whose images are contained in the smooth locus of C → T. We do not require the sections to be disjoint.

The stack 𝔐_{g,n} contains M_{g,n} as an open substack. By standard results in deformation theory, every reduced local complete intersection curve is unobstructed, and is a limit of smooth curves. Furthermore there is no obstruction to extending the sections, since these map into the smooth locus. Therefore 𝔐_{g,n} is smooth and connected, and M_{g,n} is dense in 𝔐_{g,n}. However, the stack 𝔐_{g,n} is very large (it is certainly not of finite type), and in fact it is very easy to see that its essential dimension is infinite. However, consider the open substack 𝔐_{g,n}^{fin} consisting of objects whose automorphism group is finite. Then 𝔐_{g,n}^{fin} is a Deligne–Mumford stack, and Theorem 6.1 applies to it. Thus we get the following strengthened form of Theorem 1.2 (under the assumption that 2g − 2 + n > 0).

Theorem 7.3. If 2g − 2 + n > 0 and the characteristic of k is 0, then

ed 𝔐_{g,n}^{fin} = \begin{cases}2 & if (g, n) = (1, 1), \\ 5 & if (g, n) = (2, 0), \\ 3g − 3 + n & otherwise. \end{cases}

It is not hard to show that 𝔐_{g,n}^{fin} does not have finite inertia.

8. Tate curves and the essential dimension of M_{1,0}

In this section we will finish the proof of Theorem 1.2 by showing that ed M_{1,0} = +∞.

We remark that the moduli stack M_{1,0} of genus 1 curves should not be confused with the moduli stack M_{1,1} of elliptic curves. The objects of M_{1,0} are torsors for elliptic curves, where as the objects of M_{1,1} are elliptic curves themselves. The stack M_{1,1} is Deligne–Mumford and, as we saw in the last section, its essential dimension is 2. The stack M_{1,0} is not Deligne– Mumford, and we will now show that its essential dimension is ∞.

Let R be a complete discrete valuation ring with function field K and uniformizing parameter q. For simplicity, we will assume that char K = 0. Let E = E_{q}/K denote the Tate curve over K [Sil86, §4]. This is an elliptic curve over K with the property that, for every finite field extension L/K, E(L) ≅ L^{∗}/q^{ℤ}. It follows that the kernel E[n] of multiplication by an integer n > 0 fits canonically into a short exact sequence

0 ⟶ μ_{n} ⟶ E[n] ⟶ ℤ/n ⟶ 0.

Let ∂: H^{0}(K, ℤ/n) → H^{1}(K, μ_{n}) denote the connecting homomorphism. Then it is well-known (and easy to see) that ∂(1) = q ∈ H^{1}(K, μ_{n}) ≅ K^{∗}/(K^{∗})^{n}.

Lemma 8.1. Let E = E_{q} be a Tate curve as above and let l be a prime integer not equal to char R/q. Then, for any integer n > 0,

ed E[l^{n}] = l^{n}.

Proof. We first show that ed E[l^{n}] ≥ l^{n}.

Let R′ =^{def} R[1^{1/l^{n}}] with fraction field K′ = K[1^{1/l^{n}}]. Since l is prime to the residue characteristic, R′ is a complete discrete valuation ring, and the Tate curve E_{q}/K′ is the pullback to K′ of E_{q}/K. Since ed(E_{q}/K′) ≤ ed(E_{q}/K), it suffices to prove the lemma with K′ replacing K. In other words, it suffices to prove the lemma under the assumption that K contains the l^{n}-th roots of unity.

In that case, we can pick a primitive l^{n}-th root of unity ζ and write μ_{l^{n}} = ℤ/l^{n}. Let L = K(t) and consider the class (t) ∈ H^{1}(L, μ_{l^{n}}) = L^{∗}/(L^{∗})^{n}.

It is not difficult to see that

∂_{K}(t) = q ∪ (t).

It is also not difficult to see that the order of q ∪ (t) is l^{n} (as the map α ↦ α∪(t) is injective by cohomological purity). Therefore ind(q∪(t)) = l^{n}. It follows that ind(E[l^{n}], μ_{l^{n}}) ≥ l^{n}. Then, since dim ℤ/l^{n} = 0, Corollary 4.2 implies that ed B_{k}E[l^{n}] ≥ l^{n}.

To see that ed B_{k}E[l^{n}] ≤ l^{n}, note that E[l^{n}] admits an l^{n}-dimensional generically free representation V = Ind_{μ_{l^{n}}}^{E[l^{n}]} χ, where χ: μ_{l^{n}} → 𝔾_{m} is the tautological character. Thus,

ed B_{k}E[l^{n}] dim(V) ≤ l^{n} ,

as desired; see [BR97, Theorem 3.1] or [BF03, Proposition 4.11]. ♠

Theorem 8.2. Let E = E_{q}/K denote the Tate curve over a field K as above. Then ed E = +∞.

Proof. For each prime power l^{n}, the morphism B_{k}E[l^{n}] → B_{k}E is representable of fiber dimension 1. By Theorem 3.2

ed E ≥ ed B_{k}E[l^{n}] = l^{n} − 1

for all n. ♠

Remark 8.3. It is shown in [BS08] that if A is an abelian variety over k and k is a number field then ed_{k} A = +∞. On the other hand, if k = ℂ is the field of complex numbers then ed_{ℂ}(A) = 2 dim(A); see [Bro].

Now we can complete the proof of Theorem 1.2.

Theorem 8.4. Let k be a field. Then ed_{k} M_{1,0} = +∞.

Proof. Set F = k((t)). It suffices to show that ed_{F} M_{1,0} ⊗_{k} F is infinite. Consider the morphism M_{1,0} → M_{1,1} which sends a genus 1 curve to its Jacobian. Let E denote the Tate elliptic curve over F, which is classified by a morphism Spec F → M_{1,1}. We have a Cartesian diagram:

\xymatrix{B_{k}E \ar[r] \ar[d] & M_{1,0} ⊗_{k} F \ar[d] \\ Spec F \ar[r] & M_{1,1} ⊗_{k} F.}

It follows that the morphism B_{k}E → M_{1,0} is representable, with fibers of dimension ≤ 0. Applying Theorem 3.2 once again, we see that +∞ = ed B_{F} E ≤ ed M_{1,0} ⊗_{k} F. ♠

9. On the essential dimension of the stack of principally polarized abelian varieties

Let g be a positive integer, and A_{g} be the stack of principally polarized abelian varieties of dimension g over a fixed field k of characteristic 0.This stack is well-known to be Deligne–Mumford, smooth and connected of dimension g(g + 1)/2.

The proof of the following result was suggested to us by Miles Reid for part (a), and Najmuddin Fakhruddin for part (b).

Theorem 9.1.

(a) If g is odd, then ed A_{g} = \frac{g(g+1)}{2} + 1.

(b) If g is even, write g = 2^{s}h, with h is odd. Then

\frac{g(g + 1)}{2} + 2 ≤ ed A_{g} ≤ \frac{g(g + 1)}{2} + 2^{s+3}.

Moreover, ed A_{2} = 5.

Proof. If A_{g} is the moduli space of A_{g}, by the genericity theorem we only need to know the essential dimension of the generic gerbe (A_{g})_{k(A_{g})}. Since the only non-trivial automorphism of a generic abelian variety is the antipodal map x ↦ −x, the gerbe (A_{g})_{k(A_{g})} is banded by μ_{2}, so

ed A_{g} = \frac{g(g + 1)}{2} + i_{g} ,

where i_{g} is the index of its class of this gerbe in the Brauer group of k(A_{g}).

(a) Suppose that g is odd. If g = 1 then A_{1} = M_{1,1}, and the statement follows from Theorem 1.2.

Assume g > 1. We have to show that the gerbe (A_{g})_{k(A_{g})} → Spec k(A_{g}) admits a section, for then (A_{g})_{k(A_{g})} ≃ B_{k(A_{g})}μ_{2}, and i_{g} = 1.

Each abelian scheme X → S has a canonical action of μ_{2} by the antipodal map; this embeds μ_{2} into the center of the automorphism group scheme of X → S. Let us denote by B_{g} the rigidification of A_{g} along μ_{2}, as described in [ACV03, 5.1]. There is a tautological morphism A_{g} → B_{g}. If Ω is an algebraically closed extension of k, this morphism induces an isomorphism between the isomorphism classes in B_{g}(Spec Ω) and the isomorphism classes in A_{g}(Spec Ω). If X → Spec Ω is in A_{g}(Spec Ω), the automorphism group of its image in B_{g}(Spec Ω) is the quotient Aut_{Ω}(X)/μ_{2}. The stack B_{g} is a smooth integral Deligne–Mumford stack with moduli space A_{g}.

A family X → S of principally polarized abelian varieties induces a family X/μ_{2} → S of generalized singular Kummer varieties. Consider the tautological family of principally polarized abelian varieties X_{g} → A_{g}. There is a stack X_{g}/μ_{2} → A_{g} whose objects are pairs (X → S, S → X/μ_{2}), where X → S is an object of A_{g}(S) and S → X/μ_{2} is a section of the generalized singular Kummer variety X/μ_{2} → S. Now, if (X → S, S → X/μ_{2}) is an object of X_{g}/μ_{2}, we can let μ_{2} act on it by the antipodal map on X → S, while we leave the section S → X/μ_{2} fixed; this defines an embedding of μ_{2} into the center of the automorphism group of (X → S, S → X/μ_{2}). Let W_{g} be the rigidification of X_{g}/μ_{2} along μ_{2}; there is a canonical representable morphism W_{g} → B_{g}, whose geometric fibers are generalized Kummer varieties; the zero section A_{g} → X_{g} induces a section B_{g} → W_{g}. The diagram

\xymatrix{X_{g}/μ_{2} \ar[r] \ar[d] & W_{g} \ar[d] \\ A_{g} \ar[r] & B_{g}}

is easily seen to be cartesian.

Suppose that L is an extension of k, and let X a g-dimensional abelian variety over L. Set W =^{def} X/μ_{2}, and call ρ: X → W the projection, T ⊆ W the finite subset of codimension at least 3 consisting of singular points, and w_{0} ∈ T the image of the origin in X. The étale double cover

ρ: X ∖ ρ^{−1}(T) ⟶ W ∖ T

gives rise to an equivalence of categories between μ_{2}-equivariant invertible sheaves on X and invertible sheaves on W ∖ T; the canonical sheaf ω_{W ∖T/L} corresponds to ω_{X/L}. Choose an isomorphism between ω_{X/L} and O_{X} (as sheafs of O_{X}-modules). This isomorphism becomes μ_{2}-equivariant if we let the generator −1 of μ_{2} acts on O_{X} by multiplication by (−1)^{g} = −1. This gives a μ_{2}-equivariant isomorphism of ω_{X/L}^{⊗2} with O_{X}, and an isomorphism π_{W ∖T/L}^{⊗2} ≃ O_{W ∖T}. Hence, X ∖ ρ^{−1}(T) is the double cover of square roots in ω_{W ∖T/L} of the element corresponding to 1 under the isomorphism ω_{W ∖T/L}^{⊗2} ≃ O_{W ∖T} above. Thus X can be recovered from the isomorphism ω_{W ∖T/L}^{⊗2} ≃ O_{W ∖T} as the normalization of W in the function field of’ the corresponding étale double cover.

Now, assume that W is a variety over an extension L of k, with a rational point w_{0} ∈ W (L), with the property that there exists an extension E/L and an isomorphism W_{E} =^{def} W ×_{Spec L} Spec E ≃ Y/μ_{2}, where Y is a gdimensional abelian variety over E, such that w_{0} corresponds to the image of the origin in Y/μ_{2}. Call T the singular locus of W; clearly T is finite, hence of codimension at least 3. Since H^{0}(W ∖T, O^{∗}) = L^{∗}, the Picard group of W ∖ T injects into the Picard group of (W ∖ T) ×_{Spec L} Spec E. Hence ω_{W ∖T/L}^{⊗2} is isomorphic to O_{W ∖T}. If we choose such an isomorphism, we get an étale double cover U → W ∖ T. Let X be the normalization of W ∖ T in the function field of U. We have a finite morphism X → W, which is étale over W ∖ T. The induced cover X_{E} → W_{E} is not necessarily isomorphic to Y → W_{E}, because the pullback of the isomorphism ω_{W ∖T/L}^{⊗2} ≃ O_{W ∖T} does not necessarily coincide with the isomorphism ω_{W_{E}∖T_{E}/E}^{⊗2} ≃ O_{W_{E}∖T_{E}} coming from Y. However, the obstruction to these two covers being isomorphic lives in E^{∗}/E^{∗2}, hence these covers become isomorphic after making a further, at most quadratic, extension of E. It follows that X → W is ramified over w_{0}; hence we have a unique rational point of X lying over w_{0}. So X is a variety with a rational point that becomes an abelian variety after an extension of base field, and so it is itself an abelian variety. Then it is easy to see that X/μ_{2} = W. In other words, if we have a variety with a rational point that becomes a generalized Kummer variety after a base change, it is itself a generalized Kummer variety.

Since the morphism B_{g} → A_{g} is generically an isomorphism, we have a lifting Spec k(A_{g}) → B_{g}; call w_{0} the rational point on W coming from the section B_{g} → W_{g}. Denote by W the fiber of W_{g} → B_{g}. There exists an extension E of k(A_{g}) and a lifting Spec E → A_{g} of the composite Spec E → Spec L → B_{g}. After base changing to E, the variety W becomes a generalized Kummer variety. As we have just seen, there exists an abelian variety X over k(A_{g}) such that X/μ_{2} such that X/μ_{2} = W. This variety is isomorphic over the algebraic closure \overline{k(A_{g})} to the abelian variety coming from a lifting Spec \overline{k(A_{g})} → A_{g} of morphism Spec \overline{k(A_{g})} → A_{g}; hence X acquires a principal polarization after base change to a finite extension of k(A_{g}). But since the Néron–Severi group of a generic principally polarized abelian variety is infinite cyclic, such a polarization is unique, and descends to a principal polarization over X. This yield a lifting Spec k(A_{g}) → A_{g}, and concludes the proof.

(b) Assume that g is even. We need to prove the inequalities

2 ≤ i_{g} ≤ 2^{s+3}.

Let us start from the inequality i_{g} ≥ 2: we need to show that the gerbe (A_{g})_{k(A_{g})} is not split. The restriction H_{g} → A_{g} of the Abel–Jacobi map is well known to be a locally closed embedding (the key point is that the antipodal map x ↦ −x on the Jacobian of a hyperelliptic curve is realized by the hyperelliptic involution); this induces a locally closed embedding of moduli spaces H_{g} ⊆ A_{g}. If g = 2, then this is an open embedding, so the generic gerbes of H_{2} = M_{2} and of A_{2} coincide, and i_{2} = 2, as we have seen in the proof of Theorem 7.2. This proves the equality ed A_{2} = 5.

When g ≥ 4, consider the open subscheme U ⊆ A_{g} whose geometric points correspond to abelian varieties whose automorphism group is μ_{2}. Then U is smooth, and the restriction A_{g} |_{U}→ U is a gerbe banded by μ_{2}; furthermore, since the automorphism group of a generic hyperelliptic curve is μ_{2}, we have U ∩ H_{g} ≠ ∅. Consider the class in the Brauer group of U coming from the class in H^{2}(U, μ_{2}) of A_{g} |_{U}. Since the Brauer group of U injects in the Brauer group of its field of functions k(A_{g}), if the generic gerbe of A_{g} were split, the pullback of A_{g} along any morphism Spec L → U, where L is a field, would also be split. However, the pullback of A_{g} to the generic point Spec k(H_{g}) ∈ U, which is the generic gerbe of H_{g}, is not split, as we have seen in the proof of Theorem 7.2. This gives a contradiction, and proves the inequality i_{g} ≥ 2.

Let us finish the proof by showing the inequality i_{g} ≤ 2^{s+3}. Denote by A_{g}[3] the stack whose objects are families of principally polarized abelian varieties X → S, together with a nowhere zero section S → X[3] of the 3-torsion group scheme X[3] → S, and by A_{g}[3] its moduli space. It is very well known that A_{g}[3] is integral. The natural forgetful morphism A_{g}[3] → A_{g} is a representable étale cover of degree 3^{2g} − 1; furthermore, the antipodal map x ↦ −x of an abelian variety does not leave any nonzero element of X[3] invariant. Thus A_{g}[3] is generically a scheme, and the pullback of the generic gerbe of A_{g} to Spec A_{g}[3] is split. This shows that i_{g} divides 3^{2g} − 1. To finish the proof of part (b) we need to show that the highest power of 2 that divides 3^{2g} − 1 is 2^{s+3}. Write g = 2^{s}h as in the statement of part (b). Then

3^{2g} − 1 = 3^{2^{s+1}h} − 1 = (3^{2^{s+1}} − 1) ∑_{i=0}^{h−1} 3^{2^{s+1}i} .

The second factor is odd, so we may assume that h = 1. In other words, we need to show that the highest power of 2 dividing 3^{2^{s+1}} − 1 is 2^{s+3}. This is easily done by induction on s. The base case, where s = 0 is evident, and the induction step follows from the formula

3^{2^{s+2}} − 1 = (3^{2^{s+1}} − 1)(3^{2^{s+1}} + 1) .

Here 3^{2^{s+1}} + 1 ≡ 2 (mod 4), so 3^{2^{s+1}} + 1 contributes exactly one factor of 2 to the above product. ♠

Remark 9.2. It seems natural to conjecture that i_{g} = 2, hence ed A_{g} = g(g + 1) + 2, for any even g.

Question 9.3. What is the essential dimension of the stack of (unpolarized) abelian varieties of dimension g? This is not an algebraic stack, so the techniques of this paper do not apply.

References

[ACV03] Dan Abramovich, Alessio Corti, and Angelo Vistoli, Twisted bundles and admissible covers, Comm. Algebra 31 (2003), no. 8, 3547–3618, Special issue in honor of Steven L. Kleiman.

[AV02] Dan Abramovich and Angelo Vistoli, Compactifying the space of stable maps, J. Amer. Math. Soc. 15 (2002), no. 1, 27–75.

[BF03] Grégory Berhuy and Giordano Favi, Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279–330 (electronic).

[BF04] , Essential dimension of cubics, J. Algebra 278 (2004), no. 1, 199–216.

[BR97] J. Buhler and Z. Reichstein, On the essential dimension of a finite group, Compositio Math. 106 (1997), no. 2, 159–179.

[BR05] G. Berhuy and Z. Reichstein, On the notion of canonical dimension for algebraic groups, Adv. Math. 198 (2005), no. 1, 128–171.

[Bro] P. Brosnan, The essential dimension of a g-dimensional complex abelian variety is 2g, to appear.

[BRV] Patrick Brosnan, Zinovy Reichstein, and Angelo Vistoli, Essential dimension, spinor groups and quadratic forms, http://annals.math.princeton.

edu/issues/2008/FinalFiles/BrosnanReichsteinVistoliFinal.pdf.

[BRV07]  ,  Essential dimension and algebraic stacks,  2007,

arXiv:math/0701903v1 [math.AG].

[BS08] Patrick Brosnan and Ramesh Sreekantan, Essential dimension of abelian varieties over number fields, preprint available at http://www.math.ubc.ca/

~brosnan/Papers/ab-number.pdf, 2008.

[Con] Brian Conrad, Keel–Mori theorem via stacks, http://www.math.lsa.umich.

edu/~bdconrad/papers/coarsespace.pdf.

[CS06] Vladimir Chernousov and Jean-Pierre Serre, Lower bounds for essential dimensions via orthogonal representations, J. Algebra 305 (2006), no. 2, 1055– 1070.

[CTKM06] Jean-Louis Colliot-Thélène, Nikita A. Karpenko, and Alexander S. Merkurjev, Rational surfaces and canonical dimension of PGL_{6}, to appear, 2006.

[DD99] Pierre Dèbes and Jean-Claude Douai, Gerbes and covers, Comm. Algebra 27 (1999), no. 2, 577–594. MR MR1671938 (2000a:14017)

[Del80] Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. (1980), no. 52, 137–252. MR MR601520 (83c:14017)

[DL09] Ajneet Dhillon and Nicole Lemire, Upper bounds for the essential dimension of the moduli stack of SL_{n}-bundles over a curve, 2009, preprint.

[Ful98] William Fulton, Intersection theory, second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, vol. 2, Springer-Verlag, Berlin, 1998.

[Gar] Skip Garibaldi, Cohomological invariants: exceptional groups and spin groups, to appear in Memoirs of the AMS, preprint available at arXiv:math.AG/

0411424.

[Gir71] Jean Giraud, Cohomologie non abélienne, Springer-Verlag, Berlin, 1971, Die Grundlehren der mathematischen Wissenschaften, Band 179.

[GMS03] Skip Garibaldi, Alexander Merkurjev, and Jean-Pierre Serre, Cohomological invariants in Galois cohomology, University Lecture Series, vol. 28, American Mathematical Society, Providence, RI, 2003.

[Gro64] Alexander Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I, Inst. Hautes Études Sci. Publ. Math. (1964), no. 20, 259.

[Gro67] A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. (1967), no. 32, 361.

[Har77] Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, no. 52, Springer-Verlag, New York, 1977.

[JLY02] Christian U. Jensen, Arne Ledet, and Noriko Yui, Generic polynomials, Mathematical Sciences Research Institute Publications, vol. 45, Cambridge University Press, Cambridge, 2002, Constructive aspects of the inverse Galois problem.

[KM06] Nikita A. Karpenko and Alexander S. Merkurjev, Canonical p-dimension of algebraic groups, Adv. Math. 205 (2006), no. 2, 410–433.

[KM08] , Essential dimension of finite p-groups, Invent. Math. 172 (2008), no. 3, 491–508. MR MR2393078 (2009b:12009)

[Kor00] V. È. Kordonskiĭ, On the essential dimension and Serre’s conjecture II for exceptional groups, Mat. Zametki 68 (2000), no. 4, 539–547.

[Kre99] Andrew Kresch, Cycle groups for Artin stacks, Invent. Math. 138 (1999), no. 3, 495–536.

[Led02] Arne Ledet, On the essential dimension of some semi-direct products, Canad. Math. Bull. 45 (2002), no. 3, 422–427.

[Lem04] N. Lemire, Essential dimension of algebraic groups and integral representations of Weyl groups, Transform. Groups 9 (2004), no. 4, 337–379.

[LMB00] Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 39, Springer-Verlag, Berlin, 2000.

[Mil80] James S. Milne, Étale cohomology, Princeton Mathematical Series, vol. 33, Princeton University Press, Princeton, N.J., 1980. MR MR559531 (81j:14002)

[Rei00] Z. Reichstein, On the notion of essential dimension for algebraic groups, Transform. Groups 5 (2000), no. 3, 265–304.

[RY00] Zinovy Reichstein and Boris Youssin, Essential dimensions of algebraic groups and a resolution theorem for G-varieties, Canad. J. Math. 52 (2000), no. 5, 1018–1056, With an appendix by János Kollár and Endre Szabó.

[Ser02] Jean-Pierre Serre, Galois cohomology, English ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002, Translated from the French by Patrick Ion and revised by the author. MR MR1867431 (2002i:12004)

[SGA72] Théorie des topos et cohomologie étale des schémas. Tome 2, Lecture Notes in Mathematics, Vol. 270, Springer-Verlag, Berlin, 1972, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR MR0354653 (50 #7131)

[SGA73] Groupes de monodromie en géométrie algébrique. II, Lecture Notes in Mathematics, Vol. 340, Springer-Verlag, Berlin, 1973, Séminaire de Géométrie Algébrique du Bois-Marie 1967–1969 (SGA 7 II), Dirigé par P. Deligne et N. Katz. MR MR0354657 (50 #7135)

[Sil86] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986.

(Brosnan, Reichstein) Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2

(Vistoli) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy E-mail address, Brosnan: brosnan@math.ubc.ca E-mail address, Reichstein: reichst@math.ubc.ca E-mail address, Vistoli: angelo.vistoli@sns.it