ESSENTIAL DIMENSION AND ALGEBRAIC STACKS

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

Abstract. We define and study the essential dimension of an algebraic stack. We compute the essential dimension of the stacks M_{g,n} and \overline{M}_{g,n} of smooth, or stable, n-pointed curves of genus g. We also prove a general lower bound for the essential dimension of algebraic groups with a nontrivial center. Using this, we find new exponential lower bounds for the essential dimension of spin groups and new formulas for the essential dimension of some finite p-groups. Finally, we apply the lower bound for spin groups to the theory of the Witt ring of quadratic forms over a field k.

Contents

1. Introduction 1 2. Generalities 7 3. Quotient stacks and finiteness 15 4. The essential dimension of a smooth Deligne–Mumford stack 17 5. Gerbes 22 6. Canonical dimension of smooth proper varieties 24 7. The essential dimension of a gerbe 25 8. The essential dimension of M_{g,n} for (g, n) ≠ (1, 0) 28 9. Central extensions 30 10. Tate curves and the essential dimension of M_{1,0} 31 11. Essential dimension of p-groups I 33 12. Essential dimension of p-groups II 37 13. Spinor groups 39 14. Essential dimension of cyclic p-groups 43 15. Pfister numbers 45 References 49

1. Introduction

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.

2000 Mathematics Subject Classification. Primary 14A20, 20G15, 11E04, 14H10. †Supported in part by an NSERC discovery grant. ‡Supported in part by the PRIN Project “Geometria sulle varietà algebriche”, financed

by MIUR.

Definition 1.1. Let a ∈ F (L) for L an object of Fields_{k}. A field of definition for a is an intermediate field k ⊆ K ⊆ L such that a is in the image of the induced function F (K) → F (L).

The essential dimension ed a of a (with respect to L) is the minimum of the transcendence degrees tr deg_{k} K taken over all fields of definition of a.

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}.

Note that in Definition 1.1 the essential dimension of a depends on the field L. We write ed a instead of ed(a, L) to simplify the notation.

Remark 1.2. If the functor F is limit-preserving, a condition that is satisfied in all cases that interest us, every element a ∈ F (L) has a field of definition K that is finitely generated over k, so ed a is finite. On the other hand, ed F may be infinite even in cases of interest (see for example Theorem 1.8).

Example 1.3. Let G be an algebraic group. Consider the Galois cohomology functor H^{1}(∗, G) sending K to the set H^{1}(K, G) of isomorphism classes of G-torsors over Spec(K). The essential dimension of this functor is a numerical invariant of G, usually denoted by ed G. Essential dimension was originally introduced (in [BR97, Rei00]) and has since been extensively studied in this context; see, e.g., [RY00, Kor00, Led02, JLY02, BF03, Lem04, CS06, Gar06].

Definition 1.1 is due to A. Merkurjev, as is the following observation; cf. [BF03, Proposition 1.17].

Example 1.4. Let X/k 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 ed F_{X} = dim X.

Note that the same is true if X is an algebraic space (see Proposition 2.15). Many interesting naturally arising functors are not of the form discussed

in Examples 1.3 or 1.4. One such example is the functor Curves_{g,n} that sends K into the set of isomorphism classes of n-pointed smooth algebraic curves of genus g over K. When 2g − 2 + n > 0 this functor has a wellknown extension \overline{Curves}_{g,n} that sends K into the set of isomorphism classes of n-pointed stable algebraic curves of genus g over K. Much of his paper was motivated by the following question.

Question 1.5. What are ed Curves_{g,n} and ed \overline{Curves}_{g,n}?

Our starting point is the following definition.

Definition 1.6. Suppose X is an algebraic stack over k. The essential dimension of X is the essential dimension of the functor F_{X} : Fields_{k} → Sets which sends a field L/k to the isomorphism classes of objects in X (L). We write ed X for the essential dimension of the stack X.

Note that all of the examples above may be viewed as special cases of 1.6. If X is a scheme of finite type (or an algebraic space), we recover Example 1.4. If X = BG, the classifying stack of G such that BG(T) is the category of G-torsors on T, we recover Example 1.3. Finally, Question 1.5 asks for the values of ed M_{g,n} and ed \overline{M}_{g,n}, where M_{g,n} and \overline{M}_{g,n} are the stacks of n-pointed smooth, or stable, algebraic curves of genus g over a field k.

Remark 1.7. If G is an algebraic group, we will often write ed G for ed BG. That is, we will write ed G for the essential dimension of the stack BG and not the essential dimension of the scheme underlying G. We do this to conform to the, now standard, notation described in Example 1.3. Of course, by Example 1.4, the essential dimension of the underlying scheme is dim G.

In this paper we develop the theory of essential dimension for algebraic stacks. As a first application of this theory, we give the following answer to Question 1.5.

Theorem 1.8. Assume that the characteristic of k is 0. Then

ed Curves_{g,n} = 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}.

Notice that 3g − 3 + n is the dimension of the moduli space M_{g,n} in the stable range 2g − 2 + n > 0 (or 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 stack-theoretic formalism turns out to be useful even for studying the essential dimension of algebraic groups in the classical setting of Example 1.3. Our key result in this direction is Theorem 1.10 below.

Let

(1.9)  1 ⟶ Z ⟶ G ⟶ Q ⟶ 1

denote an extension of group schemes over a field k with Z central and isomorphic to μ_{n} for some integer n > 1. For every extension K of k the sequence (1.9) induces a connecting homomorphism ∂_{K}: H^{1}(K, Q) → H^{2}(K, Z). We define 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).

(Note that ind(G, Z) does not depend on the choice of the isomorphism Z ≃ μ_{n}.)

Theorem 1.10. Let G be an extension as in (1.9). Assume that n is a prime power. Then ed G ≥ ind(G, Z) − dim Q.

Let G be a finite abstract group. We write ed_{k} G for the essential dimension of the constant group scheme G_{k} over the field k. Let exp G denote the exponent of G and let C(G) denote the center of G. One of the main consequences of Theorem 1.10 is the following result about the essential dimension of finite p-groups.

Theorem 1.11. Let G be a p-group whose commutator [G, G] is central and cyclic. Then

ed_{k} G = \sqrt{|G/C(G)|} + rank C(G) − 1 .

for any base field k of characteristic ≠ p which contains a primitive root of unity of degree exp(G).

Note that, with the above hypotheses, |G/C(G)| is a complete square. In the case where G is abelian we recover the identity ed(G) = rank(G); cf. [RY00, Example 7.4]. For most finite groups G the best previously known lower bounds on ed(G) were of the form

(1.12)  ed(G) ≥ rank(A) ,

where A was taken to be an abelian subgroup A of G of maximal rank. Theorem 1.11 represents a substantial improvement over these bounds. For example, if G is a non-abelian group of order p^{3} and k contains a primitive root of unity of degree p^{2} then Theorem 1.11 tells us that ed(G) = p, while (1.12) yields only ed(G) ≥ 2.

Theorem 1.11 has a number of interesting consequences. One of them is that ed(G) ≥ p for any non-abelian p-group G; see Corollary 12.3. Another is the following new bound on ed Spin_{n}. Here by Spin_{n} we will mean the totally split form of the the spin group in dimension n over a field k.

Theorem 1.13. Suppose k is a field of characteristic ≠ 2, and that \sqrt{−1} ∈ k. If n is not divisible by 4 then

2^{⌊(n−1)/2⌋} − \frac{n(n − 1)}{2} ≤ ed Spin_{n} ≤ 2^{⌊(n−1)/2⌋} .

If n is divisible by 4 then

2^{⌊(n−1)/2⌋} − \frac{n(n − 1)}{2} + 1 ≤ ed Spin_{n} ≤ 2^{⌊(n−1)/2⌋} + 1.

The lower bound in this theorem was surprising to us because previously the best known lower bound was the following result due of V. Chernousov and J.–P. Serre [CS06].

(1.14) ed Spin_{n} ≥ \begin{cases}⌊n/2⌋ + 1 & if n ≥ 7 and n ≡ 1, 0 or −1 (mod 8) \\ ⌊n/2⌋ & for all other n ≥ 11. \end{cases}

(The first line is due to B. Youssin and the second author in the case that char k = 0 [RY00].) Moreover, in low dimensions, M. Rost [Ros99] (cf. also [Gar06]) computed the following table of exact values:

ed Spin_{3} = 0 ed Spin_{4} = 0 ed Spin_{5} = 0 ed Spin_{6} = 0
ed Spin_{7} = 4 ed Spin_{8} = 5 ed Spin_{9} = 5 ed Spin_{10} = 4
ed Spin_{11} = 5 ed Spin_{12} = 6 ed Spin_{13} = 6 ed Spin_{14} = 7.

Taken together these results seemed to suggest that ed Spin_{n} should be a slowly increasing function of n and gave no hint of its exponential growth.

Note that the computation of ed Spin_{n} gives an example of a split, simple, connected linear algebraic group whose essential dimension exceeds its dimension. (Note that for a simple adjoint group G, ed(G) ≤ dim(G); cf. Example 13.10.) It also gives an example of a split, semi-simple, connected linear algebraic group G with a central subgroup Z such that ed G > ed G/Z. This is because ed SO_{n} = n − 1 for n ≥ 3; cf. [Rei00, Theorem 10.4].

Finally we follow a suggestion of A. Merkurjev and B. Totaro to apply our results on ed Spin_{n} to a problem in the theory of quadratic forms. Let K be a field of characteristic different from 2 containing a square root of −1, and let W(K) be the Witt ring of K. Call I(K) the augmentation ideal in W(K); it is well known that if q is a non-degenerate n-dimensional quadratic form whose class [q] in W(K) lies in I^{a}(K), then [q] can be expressed as the class a sum of a-fold Pfister forms. It is a natural to ask how many form are needed. When a = 1 or a = 2 is easy to see that n Pfister forms suffice; see Proposition 15.1. We prove the following result.

Theorem 1.15. Let k be a field of characteristic different from 2 and n an even positive integer. Then there is a field extension K/k and a class [q] ∈ I^{3}(K) represented by an n-dimensional quadratic form q/K such that [q] cannot be written as the sum of fewer than

\frac{2^{(n+4)/4} − n − 2}{7}

3-fold Pfister forms over K.

Description of contents. The rest of this paper is structured as follows.

§2 contains general results on essential dimension of algebraic stacks, which are used systematically in the rest of the paper.

§3 contains a discussion of essential dimension of quotient stacks; here we mostly rephrase known facts in our language. At the end of the section, we show finiteness of the essential dimension for a large class of algebraic stacks of finite type over a field. This class includes all Deligne–Mumford stacks and all quotient stacks of the form [X/G] for G a linear algebraic group.

In §4 we prove Theorem 4.1 about essential dimension of smooth integral Deligne–Mumford stacks satisfying an appropriate separation hypothesis; it states that the essential dimension of such a stack 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 1.8 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.

Our main result on gerbes is Theorem 5.4, stated in §5 and proved in §6 and §7. It says that the essential dimension of a gerbe banded by μ_{n}, where n is a prime power, equals the index of its class in the Brauer group. Our proof is geometric: we link the essential dimension of a gerbe banded by μ_{n} with the canonical dimension of the associated Brauer–Severi variety, and use a result of Karpenko on the canonical dimension of Brauer–Severi varieties of prime-power index.

In §8 we use Theorems 4.1 and 5.4 to compute the essential dimensions of stacks of hyperelliptic curves. This and some special arguments complete the proof of Theorem 1.8, except for the statement that ed M_{1,0} = +∞.

Theorem 5.4 is used again in §9, where we prove Theorem 1.10. The rest of the paper is dedicated to applications of this result.

In §10 we complete the proof of Theorem 1.8 by showing that ed M_{1,0} = +∞. This is achieved by applying Theorem 1.10 to the group schemes of l^{n}-torsion points on the Tate curves, where l is a prime.

§11 contains our results on p-groups. We also answer a question of Jensen, Ledet and Yui by giving an example of a finite group G with a normal subgroup N such that ed(G/N) > ed G [JLY02, p.204].

Theorem 1.13 is proved in §13, along with similar estimates for the essential dimensions of pin and half-spin groups.

Theorem 1.10 can also be applied to cyclic group over small fields. Little was known about the essential dimension of a cyclic group over ℚ until recently, when an important preprint [Flo06] of M. Florence appeared, computing the essential dimension of a cyclic group of order p^{m}, where p is a prime, over a field containing a primitive p-th root of 1; this implies that ed_{ℚ}(ℤ/p^{m}) ≥ p^{m−1}. In §14 we we recover this result as a consequence of Theorem 1.10 by making use of the Brauer-Rowen algebra, an idea we learned from [Flo06]. As a corollary of Florence’s theorem we prove a particular case of a conjecture of Ledet [Led02, Section 3], relating the essential dimensions of the cyclic group C_{n} and the dihedral group D_{n} (n odd). We show that C_{n} and D_{n} have the same essential dimension if n is a prime power and k contains a primitive p-th root of 1.

Section §15 contains our application of the results on ed Spin_{n} to the theory of quadratic forms. In particular, we prove Theorem 1.15.

Remark 1.16. One interesting issue that we do not address in this paper is the subject of the essential dimension at a prime p. For groups, this is defined in [RY00, Definition 6.3]. There is an obvious generalization to stacks which we leave to the reader to formulate. We hope that the main results of this paper would remain valid for the concept of essential dimension at a prime p (with certain obvious alterations). In particular, we think it is very likely that (in the notation of [RY00]), ed(G; p) is given by the formula in Theorem 1.11 and ed(Spin_{n}; 2) is bounded by the formulas in Theorem 1.13. However, we have not checked this in detail.

Notation. In the paper, a variety over a field k will be a geometrically integral separated scheme of finite type over k. Cohomology groups H^{i}(T, F) will be taken with respect to the fppf topology unless otherwise specified.

As explained in Remark 1.7, we will write ed G for ed BG and use these notations interchangeably. The reader may notice that we prefer to write ed BG earlier in the paper where we are working in a general stack-theoretic setting and ed G towards the end where we are primarily concerned with essential dimensions of algebraic groups.

We write μ_{n} for the groups scheme of n-th roots of unity. If k is a field, we write ζ_{n} for a primitive n-th root of unity in the algebraic closure of k. (Using the axiom of choice, we choose one once and for all.) For typographical reasons, we sometimes write C_{n} for the cyclic group ℤ/n.

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 K. Behrend, C.- L. Chai, D. Edidin, N. Fakhruddin, A. Merkurjev, B. Noohi, G. Pappas, D. Saltman and B. Totaro for very useful conversations. We would also like to thank M. Florence for sending us a copy of his preprint on the essential dimension of ℤ/p^{n} over ℚ(μ_{p}).

2. Generalities

We begin by reformulating Definition 1.1 in the language of fibered categories. For this notion, we refer the reader to the Definition 3.1 of [Vis05].

For a field k, let Points_{k} =^{def} Fields_{k} ^{op}. We study categories X which are fibered over Points_{k}; these are a generalization of functors from Points_{k} to Sets. (Clearly X is fibered over Points_{k} if and only if X is cofibered over Fields_{k} ([Gro63, 6.10]), but we prefer to work with fibered categories.)

Definition 2.1. If ξ is an object of X (K), where K is an extension of k, a field of definition of ξ is an intermediate field k ⊆ F ⊆ K, such that ξ is in the essential image of the pullback functor X (F) → X (K).

Definition 2.2. Let X be a category fibered over Points_{k}. If K is an extension of k and ξ is an object of X (K), the essential dimension of ξ, written ed ξ, is the least transcendence degree over k of a field of definition of ξ.

The essential dimension of X, denoted by ed X, is the supremum of the essential dimension of all objects ξ in X (K) for all extensions K of k.

These notions are obviously relative to the base field k. (See Remark 2.4.) We will write ed(ξ/k) (resp. ed(X /k)) when we need to be specific about the dependence on the base field.

Note that ed X takes values in the range {±∞}∪ℤ_{≥0}, with −∞ occurring if and only if X is empty.

Remark 2.3. With every functor F: Fields_{k} → Sets, one can canonically associate a category X_{F} fibered over k (see [Vis05, Proposition 3.26]). It is an easy exercise in unravelling the definitions to see that ed X_{F} as defined in 2.2 is equal to ed F as defined in 1.1.

Furthermore, given a fibered category X → Aff_{k} we get a functor

\overline{X} : Fields_{k} ⟶ Sets

sending a field K into the set of isomorphism classes in X (Spec K). It also straightforward to see that ed X equals ed \overline{X} as defined in 1.1.

Remark 2.4. Let L be an extension of a field K of transcendence degree d, and let X → Points_{L} be a fibered category. Then the composite of X → Points_{L} with the obvious functor Points_{L} → Points_{K} makes X into a category fibered over Points_{K}. We have

ed(X /K) = ed(X /L) + d.

The idea is that if F is an extension of K and ξ is an object of X (F), the image of ξ in Points_{L} defines an embedding of L in F; and every field of definition of ξ over K must contain L.

2.5. If ξ is an object in X (L) and k ⊂ F ⊂ L, then we say that ξ descends to F if F is a field of definition for ξ. The map Spec L → Spec F is then called a compression of ξ. Note that the identity morphism id_{Spec L} is always a compression.

A compression Spec L → Spec F is called a deflation if tr deg_{k} F < tr deg_{k} L. If a deflation exists, we will say that ξ is deflatable. If ξ is not deflatable, then it is called undeflatable. To show that ed X ≥ d, it suffices to exhibit an undeflatable object ξ ∈ X (L) with tr deg_{k} L = d.

We will call an undeflatable object ξ ∈ X (L) maximally undeflatable if ed ξ = ed X. Clearly, for d ∈ [0, ∞), we have ed(X /k) = d if and only if there is a maximal undeflatable object ξ ∈ X (L) with tr deg_{k} L = d.

2.6. For a scheme S, we follow Laumon & Moret-Bailly [LMB00] in letting Aff_{S} denote the category of affine schemes over S. If S = Spec k, then this category, which we denote by Aff_{k}, is equivalent to the category opposite to the category k-Alg of k-algebras. We equip Aff_{S} with the étale topology, and, by default, all notions of sheaves and stacks involving Aff_{S} are with respect to this topology.

2.7. A stack over a scheme S will mean a stack over Aff_{S}. That is, a stack over S is a category X fibered over Aff_{S} satisfying Definition 4.6 of [Vis05]. If X is a category fibered over Aff_{k}, then the restriction X̃ of X to Points_{k} via the obvious functor Points_{k} → Aff_{k} is a category fibered over Points_{k}. We write ed X =^{def} ed X̃. This defines the notion of the essential dimension of a stack.

2.8. We use [LMB00, Definition 4.1] as our definition of an algebraic stack. That is, by an algebraic stack over a scheme S, we will mean a stack X in groupoids over Aff_{S} satisfying

(1) The diagonal morphism Δ: X → X ×_{S} X is representable, separated and quasi-compact,

(2) there is an algebraic space X over S and a morphism X → X which is smooth and surjective.

We will make heavy use of the notion of gerbe. Let us recall that a category X fibered in groupoids over the category Aff_{K} is an fppf gerbe if the following conditions are satisfied.

(1) X is a stack with respect to the fppf topology.

(2) There exists a field extension K′ of K such that X (Spec K′) is not empty.

(3) Given an affine scheme S over K and two objects ξ and η in X (S), there exists an fppf cover {S_{i} → S} such that the pullbacks ξ_{S_{i}} and η_{S_{i}} are isomorphic in X (S_{i}) for all i.

A gerbe is called neutral if X (Spec K) is not empty.

We have the following easy observation.

Proposition 2.9. Let X be an algebraic stack over a field k, and let X_{red} denote the reduced substack [LMB00, Lemma 4.10]. Then ed X_{red} = ed X.

Proof. For every field K over k, the morphism X_{red} → X induces an equivalence of categories X_{red}(K) → X (K). ♠

A category X fibered over Points_{k} (resp. over Aff_{k}) is limit preserving if, whenever K = colim K_{i} is a filtered direct limit of fields (resp. k-algebras), colim X(Spec K_{i}) → X(K) is an equivalence of categories (see [Art74, p. 167]). Note that an algebraic stack (viewed as a category fibered over Aff_{k}) is limit preserving [LMB00, Proposition 4.18]. The property of being limit preserving provides the most basic instance of finiteness of essential dimension.

Proposition 2.10. If X is a limit-preserving category fibered over Points_{k}, then any object ξ of X has finite essential dimension.

Proof. Let K be a field. We can write it as a filtered direct limit K = colim_{I} K_{i} of all of its subfields K_{i} of finite transcendence degree. Since X is limit preserving, every object ξ of X(K) must be in the essential image of X (K_{i}) → X (K) for some i ∈ I. Therefore ed ξ < ∞. ♠

Let K be an extension of k; there is a tautological functor Points_{K} → Points_{k}. We denote by X_{K} the pullback of X to K. This means the following. An object of X_{K} is a pair (ξ, E), where E is an extension of K, and ξ is an object of X (E) mapping to E in Points_{k}. An arrow from (ξ, E) to (ξ′, E′) is an embedding E′ ↪ E (corresponding to an arrow E → E′ in Points_{k}) preserving K, and an arrow ξ′ → ξ in X mapping to this embedding. The category X_{K} is fibered over Points_{K}.

Proposition 2.11. Suppose that k is a field, K is an extension of k and L is an extension of K.

(a) Let X be a category fibered over Points_{k}. If ξ is an object of X (K) and ξ_{L} denotes its pullback in X (L), then

ed ξ_{L} ≤ ed ξ.

(b) Let X be a limit-preserving fibered category over Aff_{k}. If ξ is an object of X and L is contained in a purely transcendental extension of K, then

ed ξ_{L} = ed ξ.

Proof. Part (a) is obvious.

For part (b), let E be a purely transcendental extension of k containing K: then ed ξ_{E} ≤ ed ξ_{L} ≤ ed ξ by part (a), so it is enough to prove that ed ξ_{E} = ed ξ. So we can assume that L is purely transcendental over k.

We need to prove the inequality ed ξ_{L} ≥ ed ξ. Let B be a transcendence basis of L over K. Let E ⊆ L be a finitely generated subfield with tr deg_{k} E = ed ξ_{L}, with an object ξ_{E} whose image in X (L) is isomorphic to ξ_{L}; then if E ⊆ L′ ⊆ L is an intermediate field and ξ_{L′} is the image of ξ_{E} in X (L′), we have ed ξ_{L′} = ed ξ_{L}. Since L is the directed limit of subfields of the form K′(S), where K′ ⊆ K is a field of definition of ξ which is finitely generated over k and S ⊆ B is a finite subset, after enlarging L′ we can find such and L′ of the form K′(S). Let ξ_{K′} be an object of X (K′) whose image in X (K) is isomorphic to ξ. The image of ξ_{K′} in ξ_{L′} is not necessarily isomorphic to ξ_{L′}, but it will become so after enlarging K′ and S. Since we have ed ξ_{K′} ≥ ed ξ and ed ξ_{L′} = ed ξ_{L}, we may substitute K′ and L′ for K and L and assume that K and L are finitely generated over k.

We may also assume that K is infinite, because otherwise K is finite over k, we have ed ξ = 0 and the inequality is obvious. Again because X is limit-preserving, there will be an affine integral scheme U of finite type over k, with quotient field K, and an object ξ_{U} in X (U), whose image in X (K) is isomorphic to ξ. Let {x_{1}, …, x_{n}} be a transcendence basis for L over K. There will exist an open affine subscheme V of U × 𝔸_{k}^{n} and a dominant morphism V → W onto an integral affine scheme W which is of finite type of dimension ed ξ_{L} over k, together with an object ξ_{W} whose pullback in X (V) is isomorphic to the pullback of ξ_{U} along the first projection pr_{1} : W → V. Since the fraction field of U is infinite, there will exist a non-empty open subscheme U′ ⊆ U and a section U′ → W of pr_{1} : W → V. From this we we see that the restriction ξ_{U′} of ξ_{U} to U′ is isomorphic to the pullback of ξ_{W} to X (U′). If V′ denotes the closure of the image of U′ into V, we get an object ξ_{V′} of X (V′) whose image in X (U′) is isomorphic to ξ_{U′}. Hence k(V′) ⊆ k(U′) = K is a field of definition of ξ; since dim V′ ≤ dim V = ed ξ_{L} we conclude that ed ξ ≤ ed ξ_{L}. ♠

The following observation is a variant of [BF03, Proposition 1.5]. We will use it repeatedly in the sequel.

Proposition 2.12. Let X be a category fibered over Points_{k}, and let K be an extension of k. Then ed(X_{K}/K) ≤ ed(X /k).

Proof. If L/K is a field extension, then the natural morphism X_{K}(L) → X (L) is an equivalence. Suppose than M/k is a field of definition for an object ξ in X (L). Then any field N containing both M and K is a field of definition for ξ. Since there is a field N/K with tr deg_{K} N ≤ tr deg_{k} M containing both M and K (any composite of M and K), we can find a field of definition for ξ as an object in X_{K} of transcendence degree ≤ ed(X /k). ♠

Remark 2.13. The proof shows the following: if L is an extension of K and ξ is an object in some X_{K}(L), call η the image of ξ in X. Then ed ξ ≤ ed η.

In some cases of interest we can arrange for equality Proposition 2.12.

Proposition 2.14. Let X be a limit-preserving category fibered over Aff_{k}. Suppose one of the following conditions holds:

(1) k is algebraically closed.

(2) K/k is purely transcendental and k is infinite,

Then ed(X_{K}/K) = ed(X /k).

Proof. It is easy to see that, since X is limit-preserving, we can assume that K is finitely generated over k. Thus, tr deg_{k} K < ∞.

So pick ξ ∈ X (l) an undeflatable object for some field extension l/k with tr deg_{k} l = n < ∞. Again, since X is limit-preserving, we can assume that l is finitely generated over k. Set L =^{def} l ⊗_{k} K. Note that L is a field of transcendence degree n under either hypotheses (1) or (2). Write η for the restriction of ξ to L via the obvious map Spec L → Spec l. We claim that, in either case (1) or (2), η is undeflatable over K.

To show this, assume that η is deflatable over K. Then there is a intermediate field K ⊂ R ⊊ L and an object γ ∈ X (R) such that the restriction of γ to L is η. Moreover tr deg_{K} R < tr deg_{K} L. Pick affine schemes U and V of finite type over k such that k(U) = l and k(V) = K. In case (2) we can and will assume that V = 𝔸^{m}. Since X is limit-preserving, we can find an affine scheme Z of finite type over k such that k(Z) = R and γ is the restriction of some object γ̃ ∈ X (Z). Shrinking Z if necessary, we can assume that the rational map from Z to V inducing the inclusion of K into R is a morphism. We can also find a Zariski dense open W ⊂ U × V and a V -morphism f: W → Z inducing the inclusion of R into L.

For each point v ∈ V, write W_{v} (resp. Z_{v}) for the fiber of the map Z → V (resp. the map p_{2} : W → V). Since R ⊊ L, dim W > dim Z. It follows that there is a Zariski dense open subscheme M of V such that dim W_{v} > dim Z_{v} for all v ∈ M.

Suppose v is a closed point in M such that k(v) = k. Then ξ is the restriction of γ to l via the map Spec l → W_{v} → Z_{v}. It follows that ξ is deflatable over k, which is a contradiction.

Note that, in either case (1) or case (2), M(k) ≠ ∅. Therefore the contradiction is always obtained. It follows that η is undeflatable over L.

Now, we are free to pick ξ ∈ X (l) with ed(ξ/k) = ed(X /k). Then, in either case (1) or (2), we have that ed(X /k) = tr deg_{k} l = tr deg_{K} L = ed(X /L). This completes the proof of the statement. ♠

We will need the following generalization of Example 1.4.

Proposition 2.15. The essential dimension of an algebraic space locally of finite type over k equals its dimension.

Proof. Indeed, in this case X has a stratification by schemes X_{i}. Any Kpoint η: Spec K → X must land in one of the X_{i}. Thus ed X = max ed X_{i} = dim X. ♠

2.16. If X is an algebraic space over an algebraic space S, then the category of arrows T → X where T is an object in Aff_{S} is fibered over Aff_{S}. It is equivalent to the fibered category X_{h_{X}} arising from the functor h_{X}: Aff_{S} → Sets given by h_{X}(T) = Mor_{S}(T, X) via [Vis05, Proposition 3.26].

A category X fibered over Aff_{S} is said to be representable by an algebraic space if there is an algebraic space X over S and an equivalence of categories between X and X_{h_{X}}. We will follow the standard practice of identifying an algebraic space X with its corresponding representable stack X_{h_{X}}. This is permissible by Yoneda’s lemma.

A morphism f: X → Y of categories fibered over S is said to be representable if, for every algebraic space T over Y, the fiber product X ×_{Y} T is representable as a category fibered over Aff_{T}.

Let d be an integer, and let k be a field. A morphism f: X → Y of categories fibered over Aff_{k} is said to be representable of fiber dimension at most d if, for every map T → Y from an algebraic space, the fibered product X ×_{Y} T is an algebraic space locally of finite type over T with fibers of relative dimension ≤ d.

Proposition 2.17. Let X and Y be fibered categories over k. Let d be an integer, and assume that there exists a morphism X → Y that is represented by morphisms locally of finite algebraic spaces, with fiber dimension at most d. Then ed(X /k) ≤ ed(Y/k) + d.

Proof. Let K be a field over k and let x: Spec K → X be an object of X (K). Then f ∘ x: Spec K → Y is an object of Y(K), and we can find a field L with a morphism y: Spec L → Y such that k ⊂ L ⊂ K, tr deg_{k} L ≤ ed Y and the following diagram commutes.

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

Let X_{L} =^{def} X ×_{Y} Spec L. By the hypothesis, X_{L} is an algebraic space, locally of finite type over L and of relative dimension at most d. By the commutativity of the above diagram, the morphism x: Spec K → X factors through X_{L}. Let p denote the image of x is X_{L}. Since X_{L} has dimension at most d, we have tr deg_{k} k(p) ≤ d. Therefore tr deg_{k} k(p) ≤ ed Y + d. Since x factors through Spec k(p) the result follows. ♠

Remark 2.18. The proof of Proposition 2.17 clearly shows the following: For any field K/k and any ξ ∈ X (K), ed ξ ≤ ed f(ξ) + d.

The following simple observation will be used often in this paper.

Proposition 2.19. Let U be an integral algebraic space locally of finite type over k with function field K =^{def} k(U), and let f: X → U be a stack over U. Let X_{K} denote the pullback of X to Spec K. Then

ed X ≥ ed(X_{K}/K) + dim U.

Proof. If Spec L → X_{K} is maximally undeflatable over K, then the morphism Spec L → X obtained by composing with the canonical morphism X_{K} → X is maximally undeflatable over k. ♠

Let X be a locally noetherian stack over a field k with presentation P: X → X. Recall that the dimension of X at a point ξ: Spec K → X is given by dim_{x}(X) − dim_{x} P where x is an arbitrary point of X lying over ξ [LMB00, (11.14)]. Let Y be stack-theoretic closure of the image of ξ; that is, the intersection of all the closed substacks Y_{i} such that ξ^{−1}(Y_{i}) = Spec K. The morphism ξ factors uniquely through Y ⊆ X. We defined the dimension of the point ξ to be the dimension of the stack Y at the point Spec K → Y.

Proposition 2.20. Let X → Y be a morphism of algebraic stacks over a field k. Let K/k be a field extension and let y: Spec K → Y be a point of dimension d ∈ ℤ. Let X_{K} =^{def} X ×_{Y} Spec K. Then

ed(X_{K}/K) ≤ ed(X /k) − d

Proof. By [LMB00, Theorem 11.5], Y is the disjoint union of a finite family of locally closed, reduced substacks Y_{i} such that each Y_{i} is an fppf gerbe over an algebraic space X_{i} with structural morphism A_{i}: Y_{i} → Y_{i}. We can therefore replace Y by one of the Y_{i} and assume that Y is an fppf gerbe over an algebraic space Y. Without loss of generality, we can assume that Y is an integral affine scheme of finite type over k.

Let p be the image of ξ in Y. Since Y is limit-preserving, we can find an integral affine scheme U equipped with a morphism i: U → Y and a dominant morphism j: Spec K → U such that y is equivalent to i ∘ j. We can also assume that the composition U → Y → Y is dominant.

Since Y is a gerbe over Y, it follows that U → Y is representable of fiber dimension at most dim U − d. Now, form the following diagram with Cartesian squares.

\xymatrix{X_{K} \ar[r] \ar[d] & Spec K \ar[d] \\ X_{U} \ar[r] \ar[d] & U \ar[d] \\ X \ar[r] & Y}

Since the vertical maps in the lower square are representable of fiber dimension at most dim U − d,

ed(X_{K}/K) ≤ ed(X_{k(U)}/k(U))
≤ ed X_{U} − dim U
≤ ed X + dim U − d + dim U
≤ ed X − d.  ♠

In general, the inequality of Proposition 2.17 only goes in one direction. However, in important special cases we can obtain an inequality in the reverse direction.

2.21. We will say that a morphism f: X → Y of categories fibered over Points_{k} is isotropic if for every extension K of k and every object η of Y(K) there exists an object ξ of X (K) such that f(ξ) is isomorphic to η.

Proposition 2.22. Let f: X → Y be an isotropic morphism of categories fibered over Points_{k}. Then ed X ≥ ed Y.

Proof. Let K be an extension of k and η an object of Y(K). If ξ is an object of X (K) such that f(ξ) is isomorphic to η, then a field of definition for ξ is also a field of definition for η. ♠

Remark 2.23. One obvious example of an isotropic morphism is the total space of a vector bundle over a Deligne-Mumford stack. Any open substack of a vector bundle which is dense in every fiber is also isotropic.

We will use the following proposition.

Proposition 2.24. Let X and Y be categories fibered over Points_{k}. Then

ed(X ×_{Points_{k}} Y) ≤ ed X + ed Y.

Proof. This is equivalent to Lemma 1.11 of [BF03]. The proof is immediate: if (ξ, η) is an object in some (X × Y)(K), then k ⊆ F ⊆ K is a field of definition for ξ with tr deg_{k} F ≤ ed X and k ⊆ L ⊆ K is a field of definition for η with tr deg_{k} L ≤ ed Y, then the subfield of K generated by F and L is a field of definition for (ξ, η), of transcendence degree at most tr deg_{k} F + tr deg_{k} L ≤ ed X + ed Y. ♠

Remark 2.25. The inequality in Proposition 2.24 is often strict. For example, let k = ℂ, X = Bμ_{2} and Y = Bμ_{3}. Then X ×_{Points_{k}} Y = Bμ_{6}. However, we have ed μ_{n} = 1 for all integers n > 1 by [BR97, Theorem 5.3].

3. Quotient stacks and finiteness

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]. The functor F_{[X/G]} associates to a field K/k the set isomorphism classes of diagrams

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

where π is a G-torsor and ψ is a G-equivariant map.

If G is an algebraic group over k, then BG =^{def} [Spec k/G]. The functor F_{BG} is equal to the functor K ↦ H^{1}(K, G) sending K to the isomorphism classes of G-torsors over K. Remark 3.2. As noted in the introduction, for G an algebraic group, the essential dimension ed BG is equal to the essential dimension of Example 1.3 classically denoted by ed G. To prevent confusion, we remind the reader that we will use the notations ed BG and ed G interchangeably (as in Remark 1.7). Proposition 3.3. Let G → Spec K be an algebraic group acting on an algebraic space X over K and let H be a closed subgroup of G. Then

ed [X/H] ≤ ed [X/G] + dim G − dim H.

Proof. The obvious morphism [X/H] → [X/G] has fibers of dimension dim G − dim H, so this is a consequence of Proposition 2.17. ♠ Lemma 3.4. Suppose a linear algebraic group H is acting on an algebraic space X. If H is a subgroup of another linear algebraic group G then the quotient stacks [X/H] and [X ∗_{H} G/G] are isomorphic. Proof. Here X ∗_{H} G is the quotient of X × G by the H action given by h(x, g) = (xh^{−1}, hg). This fact is standard but it is as easy to prove it as it is to look for a reference.

Note that, when H acts freely on X, the quotients X/H and X ∗_{H} G/G are both algebraic spaces

Let E be an object in [X/H], i.e, an H-torsor over a k-scheme S equipped with an H-equivariant map to X. We associate to E the G-torsor E ∗_{H} G equipped with its natural morphism to the algebraic space X ∗_{H} G.

On the other hand, suppose F is an object in [X ∗_{H} G/G], i.e., a G-torsor over a k-scheme S equipped with a G-equivariant map to the algebraic space X ∗_{H} G. Consider the H-equivariant map i: X → X ∗_{H} G given by x ↦ (x, 1). We associate to F the H-torsor E over S defined by the pull-back diagram

\xymatrix{E \ar[r] \ar[d] & X \ar[d]^{i} \\ F \ar[r] & X ∗_{H} G.}

It is not difficult to see that these operations give an equivalence of categories between [X/H] and [X ∗_{H} G/G]. ♠

Now let F_{[X/G]}^{spl} be the subfunctor of F_{[X/G]} defined as follows. For any

field K/k, F_{[X/G]}^{spl}(K) consists of diagrams (3.1), where π: T → Spec(K) is a split torsor.

Following [BF04], we define the functor of orbits Orb_{X,G} by Orb_{X,G}(K) = set of G(K)-orbits in X(K).

Lemma 3.5. The functors F_{[X/G]}^{spl} and Orb_{X,G} are isomorphic.

Proof. Recall that a torsor π: T → Spec(K) is split if and only if there exists a section s: Spec(K) → T.

Now we associate the G(K)-orbit of the K-point ψs: Spec(K) → X to the object (3.1) of F_{[X/G]}^{spl}. Note that while the K-point ψs: Spec(K) → X depends on the choice of s, its G(K)-orbit does not, since any other section s′ of π can be obtained from s by translating by an element of G(K). Thus we have defined a map Orb_{X,G}(K) → F_{[X/G]}^{spl}(K) for each K/k; it is easy to see that these maps give rise to a morphism of functors

(3.6)  Orb_{X,G} ⟶ F_{[X/G]}^{spl} .

To construct the inverse map, note that a K-point p: Spec(K) → X of X, gives rise to a G-equivariant morphism ψ from the split torsor T = G × Spec(K) to X defined by ψ: (g, x) ↦ g · x. This morphism represents an object in F_{[X/G]}^{spl}(K). Translating p ∈ X(K) by g ∈ G(K) modifies ψ by composing it with an automorphism of T given by translation by g:

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

It is now easy to see that the resulting map F_{[X/G]}^{spl} → Orb_{X,G} is a morphism of functors, inverse to (3.6). ♠

Recall that a linear algebraic group G/k is called special if every G-torsor over Spec(K) is split, for every field K/k.

Corollary 3.7. Consider the action of a special linear algebraic group G/k on an algebraic space X locally of finite type k. Then

(a) The functors F_{[X/G]} and Orb_{X,G} are isomorphic.

(b) ed [X/G] ≤ dim X.

Proof. (a) Since G is special, F_{[X/G]}^{spl} = F_{[X/G]}. Now apply Lemma 3.5.

(b) Let F_{X} be the functor K → X(K). Then sending a point p ∈ X(K) to its G(K)-orbit induces a surjective morphism of functors F_{X} → Orb_{X,G}. Hence,

ed [X/G] = ed Orb_{X,G}
≤ ed F_{X}
= dim(X).  ♠

Corollary 3.8. Let G/k be a linear algebraic group and let X/k be an algebraic space, locally of finite type over k equipped with a G-action. Then ed [X/G] < ∞.

Proof. Let ρ: G → GL_{r} be an embedding and Y = X∗_{G}GL_{r}. By Lemma 3.4 the stacks [X/G] and [Y/GL_{r}] are isomorphic. Since GL_{r} is special, Corollary 3.7 tells us that ed(X/G) = ed(Y/GL_{r}) ≤ dim Y < ∞. ♠

Another consequence of Proposition 2.22 is the following “classical” theorem (see [BF03] for another proof).

Theorem 3.9. Let G be a linear algebraic group over a field k admitting a generically free representation on a vector space V. Then

ed BG ≤ dim V − dim G.

Proof. Let U denote a dense G-stable Zariski open subscheme of V on which G acts freely. Then [U/G] is an algebraic space of dimension dim V − dim G and the map [U/G] → BG is representable and isotropic. ♠

Finiteness. The main theorem on finiteness of essential dimension is now an easy corollary our study of quotient stacks and of a result of A. Kresch.

Theorem 3.10. Let X be an algebraic stack of finite type over k. If 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(X /k) < ∞.

Proof. By a Theorem of Kresch [Kre99, Proposition 3.5.9] X is covered by quotient stacks [X_{i}/G_{i}]. By Corollary 3.8, ed X = max_{i} ed [X_{i}/G_{i}] < ∞. ♠

Theorem 3.10 does not hold without the assumption that all the \underline{Aut}_{Ω}(ξ) are affine. For example, by Theorem 1.8, ed M_{1,0} = +∞. The proof of this will be given in §10, and we will also see (Theorem 10.2) that ed BE = +∞ if E is the Tate elliptic curve over the power series field ℂ((t)).

4. The essential dimension of a smooth Deligne–Mumford stack

The goal of this section is to prove the following theorem which allows us, in several of the most interesting cases, to reduce the calculation of the essential dimension of a stack to that of the essential dimension of a gerbe over a field.

Recall that if X is an algebraic stack over a base scheme S, then X is said to be separated over S when the diagonal morphism Δ : X → X ×_{S} X is proper. In the case of a Deligne–Mumford stack, the diagonal morphism is always quasi-finite. So the diagonal morphism of a separated Deligne– Mumford stack is finite.

Recall also that the inertia stack I_{X} → X is the fibered product

X ×_{X ×X} X

mapping to X via the second projection (with both maps X → X × X given by the diagonal). The inertia stack is a group stack, and represents the functors of isomorphisms of objects: that is, it is equivalent to the obvious fibered category over S whose objects are pairs (ξ, α), where ξ is an object over some morphism T → S, and α is an automorphism of ξ in X (T).

We say that X has finite inertia when I_{X} is finite over X. A separated Deligne–Mumford stack has finite inertia; however, having finite inertia is a weaker condition than being separated. For example, when X is a scheme the inertia stack is the identity X = X, so X always has finite inertia, even when it is not separated.

By a result of Keel and Mori ([KM97], see also [Con]) an algebraic stack locally of finite type over Spec k with finite inertia has a moduli algebraic space X, which is also locally of finite type over Spec k. The morphism X → X is proper.

Theorem 4.1. Let k be a field of characteristic 0, X a smooth connected Deligne–Mumford stack with finite inertia, locally of finite type over Spec k. Let X the moduli space of X, K the field of rational functions on X. Denote by X_{K} the fibered product Spec K ×_{X} X. Then we have

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

Corollary 4.2. If X is as above and U is an open dense substack, then ed(M/k) = ed(U/k).

Corollary 4.3. If the conditions of the theorem are satisfied, and the generic object of X has no non-trivial automorphisms (X is an orbifold, in the topologists’ terminology), then ed(X /k) = dim X.

Corollary 4.4. Assume that k has characteristic 0. If g ≥ 3, or g = 2 and n ≥ 1, or g = 1 and n ≥ 2, then

ed(M_{g,n}/k) = ed( \overline{M}_{g,n}/k) = 3g − 3 + n.

Proof. In all these case 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}. Similarly for \overline{M}_{g,n}. ♠

Proof of Theorem 4.1. The equality ed(X /k) ≥ dim X+ed(X_{K}/K) is clear. Let us prove the opposite inequality.

Let F be an extension of k and ξ an object in X (F), corresponding to a morphism ξ: Spec F → X. We need to show that the essential dimension of ξ is less than or equal to dim X + ed(X_{K}/K). Of course we can assume that F is infinite, otherwise ed ξ would be 0, in which case we are done.

We may also assume that X is an affine scheme. If it is not so, by [Knu71,

II, Theorem 6.4] the composite Spec F ⟶^{ξ} X → X admits a factorization Spec F → U → X, where U is an affine scheme and the morphism U → X is étale. By substituting X with the pullback Spec U ×_{X} X the dimension stays the same, while the essential dimension of the generic gerbe can not increase.

We proceed by induction on the codimension in X of the closure of the image of the composite Spec F ⟶^{ξ} X → X. If this codimension is 0, then ξ: Spec F → X factors though X_{K}, in which case the inequality is obvious. So we can assume that this codimension is positive, that is, the composite Spec F ⟶^{ξ} X → X is not dominant.

Claim 4.5. There a morphism Spec F [[t]] → X, such that its restriction Spec F ⊆ Spec F [[t]] → X is isomorphic to ξ, and such that the image of the composite Spec F [[t]] → X → X consists of two distinct points.

By Schlessinger’s theorem, there exists a local complete noetherian F - algebra A with residue field F and is a formal versal deformation of ξ defined on A. Since X is not obstructed the ring A is a power series ring F [[t_{1}, …, t_{m}]]. The composite Spec A → X → X is dominant; since F is infinite, if a_{1}, …, a_{m} are general elements of F and the homomorphism A → F [[t]] is defined by sending t_{i} to a_{i}t, the composite Spec F [[t]] → Spec A → X has the required properties.

Claim 4.6. There exists a complete discrete valuation subring R ⊆ F [[t]] and fraction field L ⊆ F ((t)), such that the following properties hold:

(a) t ∈ R,

(b) the residue field of R has transcendence degree over k at most equal to dim X + ed(X_{K}/K), and

(c) The composite morphism Spec F ((t)) ⊆ Spec F [[t]] → X factors through Spec L.

Consider the composite Spec F ((t)) ⊆ Spec F [[t]] → X. The closure of its image in X has a codimension that is less than the codimension of the closure of Spec F; hence by induction hypothesis there exists an intermediate field k ⊆ E ⊆ F ((t)) such that Spec F ((t)) → X factors through Spec E, and tr deg_{k} L is at most dim X + ed(X_{K}/K).

Set L =^{def} E(t) and R =^{def} L ∩ F [[t]]; clearly R is a discrete valuation subring in L containing t. We claim that the transcendence degree of the residue field R/𝔪_{R} over k is less than

tr deg_{k} L ≤ dim X + ed(X_{K}/K) + 1.

This is elementary: if s_{1}, …, s_{r} are elements of R whose images in R/𝔪_{R} are algebraically independent over k, then it is easy to check that s_{1}, …, s_{r}, t are algebraically independent over k. Thus R and L satisfy all the conditions of the claim, except completeness. By completing R we prove the claim.

Claim 4.7. The morphism Spec F [[t]] → X factors through Spec R.

This claim implies that ξ: Spec F → X factors through Spec(R/𝔪_{R}), which shows that

ed ξ ≤ tr deg((R/𝔪_{R})/k)
≤ dim X + ed(X_{K}/K),

thus proving the Theorem.

To prove the claim, let us first show that the morphism Spec F [[t]] → X factors through Spec R. This is trivial, since X is an affine scheme: if X = Spec A, the homomorphism A → F ((t)) corresponding to the composite Spec F ((t)) ⊆ Spec F [[t]] → X → X factors through F [[t]] and also through L, so its image in contained in L ∩ F [[t]] = R.

Now denote by X_{R} the normalization of the reduced pullback (Spec R ×_{X} X)_{red}; by a well known theorem of Nagata, stating that the normalization of a complete local integral domain is finite, this is finite over (Spec R×_{X}X)_{red}. The restriction of X_{R} to Spec L ⊆ Spec R coincides with (Spec L ×_{X} X)_{red}; hence the morphism Spec L → X yields a morphism Spec L → X_{R}. Thus the moduli space X_{R} of X_{R}, which is integral and finite over Spec R, admits a section over Spec L; hence X_{R} = Spec R.

Since Spec F [[t]] is normal and the morphism Spec F [[t]] → (Spec R ×_{X} X)_{red} induced by Spec F [[t]] → X is dominant, the morphism Spec F [[t]] → X factors through X_{R}.

Let X_{0} → X_{R} be an étale map, where X_{0} is a scheme. Since R is complete, hence henselian, X_{0} contains a component of the form Spec R_{0}, where R_{0} is a discrete valuation ring which is a finite extension of R. This component dominates X_{R}, so we can assume X_{0} = Spec R_{0}. Set X_{1} = X_{0} ×_{X_{R}} X_{0}; we have that X_{1} = Spec R_{1}, where R_{1} is a product of discrete valuation rings. The stack X_{R} has a presentation X_{1}⇉X_{0}. If we set Spec F [[t]]×_{X_{R}} Spec R_{0} = Spec A, we have that A is a product of discrete valuation rings, each of them an étale extension of F [[t]]; this implies that the image of t ∈ R in A is a uniformizing parameter in all of the factors of A; and this implies that t is also a uniformizing parameter in R_{0}. So R_{0} is étale over R, because the characteristic of the base field is 0.

Now consider the morphism Spec L → X_{R}: set Spec L_{0} = Spec L ×_{X_{R}} X_{0} and Spec L_{1} = Spec L ×_{X_{R}} X_{1} = Spec(L_{0} ⊗_{L} L_{0}). Set also S_{i} = R_{i} ⊗_{R} F [[t]] and M_{i} = L_{i} ⊗_{L} F ((t)) for i = 0 or 1. The ring M_{i} is a product of fields, S_{i} a product of discrete valuation rings.

The pullback of the commutative diagram

\xymatrix{Spec L_{1} \ar[r] \ar[r] \ar[d] & Spec L_{0} \ar[r] \ar[d] & Spec L \ar[d] \ar[dr] \\ Spec R_{1} \ar[r] \ar[r] & Spec R_{0} \ar[r] & X_{R} \ar[r] & Spec R}

via the morphism Spec F [[t]] → Spec R is isomorphic to the diagram

\xymatrix{Spec M_{1} \ar[r] \ar[r] \ar[d] & Spec M_{0} \ar[r] \ar[d] & Spec F ((t)) \ar[d] \ar[dr] \\ Spec S_{1} \ar[r] \ar[r] & Spec S_{0} \ar[r] & X_{F [[t]]} \ar[r] & Spec F [[t]].}

But the morphism Spec F ((t)) → X_{F [[t]]} extends to a morphism Spec F [[t]] → X_{F [[t]]}; and this implies that M_{0} is unramified over F ((t)) with respect to the canonical valuation of F ((t)). Hence L_{0} is unramified over L; if we denote by T_{i} the normalization of R_{i} in L_{i} we have that T_{0} is étale over R, and T_{1} = T_{0} ⊗_{R} T_{0}. The diagram

\xymatrix{Spec L_{1} \ar[r] \ar[r] \ar[d] & Spec L_{0} \ar[d] \\ Spec R_{1} \ar[r] \ar[r] & Spec R_{0}}

extends to a diagram

\xymatrix{Spec T_{1} \ar[r] \ar[r] \ar[d] & Spec T_{0} \ar[d] \\ Spec R_{1} \ar[r] \ar[r] & Spec R_{0}}

which defines the descent data for a morphism Spec R → X_{R} extending Spec L → X_{R}. This proves the theorem. ♠

Remark 4.8. The stack X_{K} that appears in the statement of Theorem 4.1 can be defined in much greater generality. Let X be a locally noetherian integral algebraic stack. It is easy to see that all dominant maps Spec K → X are equivalent, in the sense of [LMB00, Definition 5.2], hence they define a point of X, called the generic point of X. Then the stack X_{K} of Theorem 4.1 is the gerbe of X at its generic point, in the sense of [LMB00, §11.1]. This is naturally called the generic gerbe of X.

Theorem 4.1 is false in general for algebraic stacks, and also for singular Deligne–Mumford stacks.

Examples 4.9.

(a) Let k be any field. Let G =^{def} 𝔾_{a} × 𝔾_{a} act on 𝔸^{3} be 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} × BG; 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] ≃ Bμ_{r}^{n}, which has essential dimension n.

(c) The following example shows that Corollary 4.2 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 point of [X/G]) = 0.

On the other hand, let Y be the locus of matrices of rank n − 1, which forms 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 an isotropic morphism [Y/G] → ℙ^{n−1}. Hence by Proposition 2.19 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 4.10. One could ask to what class of curves we may apply Theorem 4.1. More specifically, let 𝔐_{g} be the stack over Spec k, whose objects over a k-scheme S are flat proper finitely presented maps C → S, whose geometric fibers are reduced locally complete intersection irreducible curves of geometric genus g. The stack 𝔐_{g} is an irreducible locally finitely presented smooth stack over Spec k, but it is not Deligne–Mumford. It is not of finite type, either, and it is easy to see that ed 𝔐_{g} = ∞ for all g.

However, 𝔐_{g} will contain the largest open substack 𝔐̃_{g} with finite inertia: it follows from Theorem 4.1 that ed 𝔐̃_{g} = ed M_{g}. Is there a good description of 𝔐̃_{g}? Does it contain all curves with finite automorphism groups?

5. Gerbes

In this section, we address the problem of computing the essential dimension a gerbe over a field K. The general problem seems difficult; however we do have a formula in the case where the gerbe is banded by μ_{p^{n}} for p a prime.

Let G be a sheaf of abelian groups in the category Aff_{K}. A gerbe X over K is said to be banded by G if for any affine K-scheme S and any object ξ of X there is an isomorphism of group schemes G_{S} ≃ \underline{Aut}_{S}(ξ), which is compatible with pullbacks, in the obvious sense. (Here \underline{Aut}_{S}(ξ) denotes the group scheme of automorphisms of ξ over S.) A gerbe banded by G is neutral if and only if it is equivalent to the classifying stack B_{K}G.

More generally, [Gir71] contains a notion of gerbe banded by G when G is not abelian; but we do not need the added generality, which makes the definition considerably more involved.

There is a natural notion of equivalence of gerbes banded by G; the set of equivalence classes is in natural bijective correspondence with the group H^{2}(K, G). The identity is the class of the neutral gerbe B_{K}G.

5.1. 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 (CSAs) over K. By Wedderburn’s structure theorem, any CSA over K isomorphic to the matrix algebra M_{n}(D) for D a division algebra over K which is unique up to isomorphism. Moreover, if A and B are two Brauer equivalent CSAs, the division algebras D and E corresponding to A and B respectively are isomorphic. For a class [A] ∈ Br(K), the index of A is \sqrt{dim_{K} D}.

Let n denote a non-negative integer and α ∈ H^{2}(K, μ_{n}). We define the index ind α to be the index of the image on α under the composition

H^{2}(K, μ_{n}) ⟶ H^{2}(K, 𝔾_{m}) ⟶^{≅} Br(K).

Note that the index of α is the smallest integer d such that α is in the image of the (injective) connecting homomorphism

(5.2)  ∂: H^{1}(K, PGL_{d}) ⟶ H^{2}(K, μ_{d})

arising from the short-exact sequence

(5.3)  1 ⟶ μ_{d} ⟶ SL_{d} ⟶ PGL_{d} ⟶ 1.

The exponent ord([A]) of a class [A] ∈ Br K is defined to be its order in the Brauer group. Note also that the exponent ord([A]) always divides the index ind([A]) [Her68, Theorem 4.4.5].

The next two sections will be devoted to the proof of the following result.

Theorem 5.4. Let X be a gerbe over a field K banded by μ_{p^{m}} for p a prime and k a positive integer. Then

ed X = ind [X].

6. Canonical dimension of smooth proper varieties

The following lemma is well known.

Lemma 6.1. Let X, Y and Z be varieties over K. Assume that Y is smooth and Z is proper. If there exist rational maps α: X −−→ Y and β: Y −−→ Z, then there exist a rational map γ: X −−→ Z.

Proof. Immediate from Nishimura’s lemma; cf. [Nis55] or [RY00, Proposition A.6]. ♠

Definition 6.2. Two smooth proper varieties X and Y are e-equivalent (or simply equivalent) if there exist rational maps X −−→ Y and Y −−→ X.

From Lemma 6.1 above, this is in fact an equivalence relation.

Definition 6.3. If X and Y are smooth proper varieties over K, let e(X, Y) denote the least dimension of the closure of the image of a rational map X −−→ Y. We set e(X, Y) = +∞ if there are no rational maps X −−→ Y and define e(X) = e(X, X).

The integer e(X) has been introduced in [KM06], and is called the canonical dimension of X. (In the case where X is a Brauer-Severi variety of dimension n − 1, this number coincides with the canonical dimension of the class of X in H^{1}(K, PGL_{n}), as defined in [BR05].)

Lemma 6.4. Let X, X′, Y and Y′ be smooth proper varieties over K, such that X is equivalent to X′ and Y is equivalent to Y′. Then e(X, Y) = e(X′, Y′).

Proof. Let f: X −−→ Y be a rational map such that dim V = e(X, Y), where V ⊆ Y is the closure of the image of f. From Lemma 6.1, there exists a rational map X′ −−→ V; by composing this with the embedding V ⊆ Y, we see that there a rational map X′ −−→ Y whose image has dimension at most dim V = e(X, Y). Hence e(X′, Y) ≤ e(X, Y).

On the other hand, from the same lemma we see that there a rational map V → Y′; this can be composed with the dominant rational map X −−→ V to obtain a rational map X → Y′ whose image has dimension at most dim V = e(X, Y). Hence e(X, Y′) ≤ e(X, Y). We deduce that

e(X′, Y′) ≤ e(X′, Y) ≤ e(X, Y);

by symmetry, e(X, Y) = e(X′, Y′). ♠

Corollary 6.5. If X and Y are equivalent smooth proper varieties over K, then

e(X) = e(Y) = e(X, Y).

When we have resolution of singularities for varieties over K, then e(X) can also be defined as the least dimension of a smooth proper variety in the equivalence class of X.

7. The essential dimension of a gerbe

Let K be a field and let n be an integer with n > 1. Let X → Spec K be a gerbe banded by μ_{n} with index d. Let P → Spec K be a Brauer– Severi variety of dimension d − 1 whose class in H^{1}(K, PGL_{d}) maps to [X] ∈ H^{2}(K, μ_{n}) ⊆ Br K under the connecting homomorphism H^{1}(K, PGL_{d}) → Br K.

Theorem 7.1. ed X = e(P) + 1.

Before proving the theorem we will prove the following easy corollaries.

Corollary 7.2. If X_{1} and X_{2} are gerbes over Spec K banded by μ_{n_{1}} and μ_{n_{2}} respectively whose cohomology classes in Br K are the same, then ed X_{1} = ed X_{2}.

Proof. Clear. ♠

Corollary 7.3. Theorem 5.4 holds. That is, if n = p^{m}, with m > 0, then ed X = ind [X].

Proof. If the index d is 1, then X is neutral. Thus X = Bμ_{n} with n > 1. By [BF03, Example 2.3], ed Bμ_{n} = 1.

Assume then that d > 1. Then the class of X in Br K is also represented by a gerbe banded by μ_{d}. By Corollary 7.2, we can substitute this gerbe for X, and assume d = n.

The class of X in H^{2}(K, μ_{d}) comes from a division algebra of degree d. If P is the associated Brauer–Severi variety then e(P) = d − 1 by a theorem of Karpenko [Kar00, Theorem 2.1] (see also [Mer03, §7.2]). ♠

Proof of Theorem 7.1. Since the exponent of ∂P divides n, there exists an invertible sheaf Λ on P whose degree when base changed to P\overline{K} ≅ ℙd−1 ; \overline{K} is n.

For each K scheme T, we denote by Λ_{T} the pullback of Λ to P_{T} =^{def} P ×_{K} T.

The gerbe X is equivalent to the gerbe whose sections over a K-scheme T consist of pairs (L, λ) where L is a an invertible sheaf on P_{T}, and λ is an isomorphism of sheaves of O_{P_{T}} -modules between L^{⊗n} and Λ_{T}. We may, therefore, substitute this gerbe for X in proving Theorem 7.1.

Let P^{∨} denote the dual Brauer–Severi variety. Since P splits over k(P^{∨}) and P^{∨} splits over K(P), P and P^{∨} are in the same e-equivalence class. Thus, e(P) = e(P^{∨}). Each point ξ ∈ P^{∨} gives, by definition, a hypersurface of degree 1 in P_{k(ξ)}, which we denote by H_{ξ}.

Claim. ed X ≤ e(P) + 1.

Proof. Let F be an extension of K, and let (L, λ) be a class in X (Spec F). The degree of the pullback of L to P\overline{F} ≅ ℙd−1 ; \overline{F} is 1. So H^{0}(P_{F} , L) is an n-dimensional vector space over F. Choose a non-zero section of L and let H ⊂ P_{F} denote its divisor. Then H gives a morphism Spec F → P^{∨}. We know that there exists a rational map P^{∨} −−→ P^{∨} whose image has dimension e(P^{∨}) = e(P); call all V the closure of its image. Since there is a rational map P^{∨} −−→ V and P^{∨}(F) ≠ ∅, we also have V (F) ≠ ∅. Chose a morphism Spec F → P^{∨} whose image is contained in V, and call its image ξ. The transcendence degree of K(ξ) over K is at most e(P). Then k(ξ) ⊆ F and the pullback of O_{P_{k(ξ)}}(H_{ξ}) to P_{F} is O_{P_{F}} (H) ≃ L. Fix an isomorphism of invertible sheaves μ: O_{P_{k(ξ)}}(H_{ξ})^{⊗n} ≃ Λ_{k(ξ)}: the pullback of μ gives an isomorphism of invertible sheaves L^{⊗n} ≃ Λ_{F}, which will differ from λ by an element a ∈ F^{∗}. Then (L, λ) is clearly defined over the field k(ξ)(a), whose transcendence degree over K is at most e(P) + 1. ♠

Now we prove that ed X ≥ e(P) + 1.

Let S ⊆ P be a divisor in the linear system of Λ; when pulled back to P\overline{K} ≃ ℙd−1 ; \overline{K}, the hypersurface S\overline{K} has degree n. If F is an extension of K, we can determine an element of X (Spec F) by specifying a hyperplane H ⊆ P_{F} and a rational function u ∈ k(P_{F}) whose divisor is S_{F} − nH (here, as in what follows, we will write S_{F} to indicate the pullback of S to P_{F}): the line bundle is O_{P_{F}} (H), and the isomorphism

O_{P_{F}} (H)^{⊗n} = O_{P_{F}} (nH) ≃ O_{P_{F}} (S_{F}) = Λ_{F}

is given by multiplication by u. Every element of X (Spec F) is isomorphic to one arising in this way. The hyperplane H and the rational function u are not unique, but it is easy to see that the class of u in k(P_{F})^{∗}/k(P_{F})^{∗n} is uniquely determined by the element of X (Spec F). This gives us an invariant, which is functorial in F.

Let P^{∨} −−→ P^{∨} be a rational function whose image has dimension e(P), and call V the closure of its image in P^{∨}. The generic point Spec k(V) → V ⊆ P^{∨} gives us a rational point ξ of P_{k(V)}^{∨}, corresponding to a hyperplane H_{ξ} ⊆ P_{k(V)}; let u be a rational function on P_{k(V)} whose divisor is S_{k(V)} − nH_{ξ}. Consider the element α ∈ X (Spec k(P)(t)) determined by the rational function tu, whose divisor is S_{k(V)(t)} − nH_{ξ}. We claim that α can not come from an extension of K whose transcendence degree is less than e(P) + 1.

Let F be a subfield of k(V)(t) containing K such that α is defined over F; we need to show that the transcendence degree of F over K is e(P) + 1. Let Z be an integral variety over K with quotient field F. Since X has a section on F = k(Z) there exists a hyperplane H ⊆ P_{k(Z)}, which gives a rational map ψ: Z −−→ P^{∨}. By blowing up the base locus we can assume that ψ is a morphism.

Consider the rational map ϕ: V ×_{K} ℙ^{1} −−→ Z corresponding to the embedding k(Z) ⊆ k(V)(t). The class of tu in k(P_{k(V)(t)})^{∗}/k(P_{k(V)(t)})^{∗n} comes from k(P_{k(Z)})^{∗}/k(P_{k(Z)})^{∗n}; hence there exist rational functions v ∈ k(P_{k(Z)})^{∗} and w ∈ k(V_{k(P)(t)})^{∗} such that

tu = w^{n}(id_{P} × ϕ)^{∗}v ∈ k(P_{k(V)(t)}) = k(P × V × ℙ^{1}).

Now, the valuation of tu along the divisor D =^{def} P × V × {0} ⊆ P × V × ℙ^{1} is 1; since n > 1, the valuation of (id_{P} × ϕ)^{∗}v at D can not be 0. Hence D can not dominate P × Z under the map id_{P} × ϕ, or, equivalently, V × {0} can not dominate Z under ϕ. The restriction of ϕ gives a regular function V = V × {0} → Z. The composite

V ⟶^{ϕ} Z ⟶^{ψ} P^{∨},

which is well defined because ψ is a morphism, has image closure of dimension less than dim Z: hence, if dim Z ≤ e(P) = e(P^{∨}), by composing with the dominant rational map P^{∨} −−→ V we obtain a rational map P^{∨} −−→ P^{∨} with image closure less than e(P^{∨}), a contradiction. So the dimension of Z, which equals the transcendence degree of F over K, is e(P) + 1, as claimed.

This completes the proof of Theorem 7.1 and, thus, the proof of Theorem 5.4. ♠

When the index of P is not a prime power, the essential dimension of P is smaller than the index. In fact, let m be the index of P, and consider the prime decomposition m = p_{1}^{a_{1}} … p_{r}^{a_{r}}. Then the class of P in Br K is the product of the classes α_{1}, …, α_{r} of indices p_{1}^{a_{1}}, …, p_{r}^{a_{r}}. If P_{1}, …, P_{r} are Brauer–Severi varieties with classes α_{1}, …, α_{r}, then the splitting fields of P are exactly the fields that split all of the P_{i}; hence P_{1} × ⋯ × P_{r} has a point over k(P), and P has a point over k(P_{1} × ⋯ × P_{r}). So P and P_{1} × ⋯ × P_{r} are equivalent, and we have

e(P) = e(P_{1} × ⋯ × P_{r})
≤ p_{1}^{a_{1}} + ⋯ + p_{r}^{a_{r}} − r.

In [CTKM06], Colliot-Thélène, Karpenko and Merkurjev conjecture that equality always holds and prove it for m = 6. This can be reformulated in the language of essential dimension of gerbes in the following fashion.

Conjecture 7.4. If X is a gerbe banded by μ_{n} over a field K, let p_{1}^{a_{1}} … p_{r}^{a_{r}} be the decomposition into prime factors of the index of the class of X in the Brauer group of K. Then

ed X = p_{1}^{a_{1}} + ⋯ + p_{r}^{a_{r}} − r + 1.

When the index is 6 this follows from [CTKM06, Theorem 1.3].

In view of the fact that this holds for r = 1, the conjecture can be rephrased as follows: if m and n are relatively prime positive integers, X and Y are gerbes banded by μ_{m} and μ_{n}, then

ed(X × Y) = ed X + ed Y − 1.

Or, back to the language of canonical dimension, one could ask the following more general question. Let X and Y be smooth projective varieties over a field K. Assume that there are no rational functions X −−→ Y or Y −−→ X. Then is it true that e(X × Y) = e(X) + e(Y)? A positive answer to this question would imply the conjecture above.

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

In this section we complete the proof of Theorem 1.8 when (g, n) ≠ (1, 0). By Corollary 4.4, it will suffice to compute ed M_{0,0}, ed M_{0,1} and ed M_{0,2}, ed M_{1,1} and ed M_{2,0}.

It is easy to see that ed M_{0,1} = ed M_{0,2} = 0. Indeed, 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. Alternatively, M_{0,2} = B𝔾_{m} and M_{0,1} = B(𝔾_{m} ⋉ 𝔾_{a}), and the groups 𝔾_{m} and 𝔾_{m} ⋉ 𝔾_{a} are special (and hence have essential dimension 0).

We will now consider the remaining cases, starting with M_{0,0}.

Since M_{0,0} ≃ BPGL_{2}, the fact that ed M_{0,0} = 2 is classical. We recall the following result.

Proposition 8.1. Let k be an algebraically closed field of characteristic 0. Then ed PGL_{n} = 2 for n = 2, 3 and 6.

This is [Rei00, Lemma 9.4 (c)]. However, note that the proof does not really require the field k to be algebraically closed of characteristic 0: it goes through whenever char k does not divide n and k contains a primitive n-th root of unity. Thus we have the following.

Corollary 8.2. Let k be a field of characteristic not equal to 2. Then ed M_{0,0} = ed BPGL_{2} = 2.

This can also be proved directly very easily: 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.

We will now proceed to compute the essential dimension of M_{1,1}.

Proposition 8.3. If k is a field of characteristic not equal to 2 or 3, then

ed M_{1,1} = 2.

Proof. Every elliptic curves over a field K can be written as a cubic in ℙ_{K}^{2} with equation yz = x^{3} + axz^{2} + bz^{3}, so it is defined over k(a, b). Hence ed M_{1,1} ≤ 2.

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 neutral by [Sil86, Proposition 1.4 (c)], and so ed X = ed B_{k(j)}μ_{2} = 1. This implies what we want. ♠

Ir remains to compute the essential dimension of M_{2,0}. The equality ed M_{2,0} = 5 is a special case of the following more general result.

Theorem 8.4. Let H_{g} denote the stack of hyperelliptic curves of genus g > 1 over a field k of characteristic 0 and let \overline{H}_{g} denote its closure in \overline{M}_{g}.

Then

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

Since H_{2} = M_{2,0} and \overline{H}_{g} = \overline{M}_{2}, Theorem 8.4 completes the proof of every case of Theorem 1.8, except for the fact that ed M_{1,0} = +∞.

Proof. The closure \overline{H}_{g} is well known to be smooth, so by Corollary 4.2 it is enough to prove the statement about H_{g}. 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 4.1 we have

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

so we need to show that ed((H_{g})_{K}/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.

Call D_{g} the stack over K whose object 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 which 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). Call (P → Spec K, Δ) object of D_{g}(Spec K) corresponding the generic point Spec K → H_{g}. It is well known, and easy to show, that P (K) = ∅. 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 μ_{2} 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 5.4 its essential dimension is the index of [P], which equals 2. ♠

9. Central extensions

The rest of this paper will rely on our analysis of the following situation which we recall from the introduction (1.9).

Let

(9.1)  1 ⟶ Z ⟶ G ⟶ Q ⟶ 1

denote an extension of group schemes over a field k with Z central and isomorphic to μ_{n} for some integer n > 1. As in the introduction, we define 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).

We are now going to prove Theorem 1.10 from the introduction which we restate for the convenience of the reader.

Theorem 9.2. Let G be an extension as in (9.1). Assume that n is a prime power. Then

ed(BG/k) ≥ ind(G, Z) − dim Q.

Proof. Let K/k be a field extension and let t : Spec K → BQ be a Q-torsor over Spec K. The dimension of BQ at the point t is − dim Q. Let X denote the pull-back in the following diagram.

\xymatrix{X \ar[r] \ar[d] & Spec K \ar[d]^{t} \\ BG \ar[r] & BQ}

By Proposition 2.20, ed(X /K) ≤ ed(BG/k) + dim Q. On the other hand, since BG is a gerbe banded by Z over BQ, X is a gerbe banded by Z over Spec K. Therefore, by Theorem 5.4, ed(X /K) = ind ∂_{K}(t). By substitution, ind ∂_{K}(t) − dim Q ≤ ed(BG/k). Since this inequality holds for all field extensions K/k and all Q-torsors t over K, the result follows. ♠

Remark 9.3. An affirmative answer to Conjecture 7.4 would yield an inequality similar to the one in 9.2 without the assumption that n is a prime power: Let ind(G, Z) = ∏ p_{i}^{a_{i}} be the prime factorization of ind(G, Z). The conjecture would imply that

ed(BG/k) ≥ 1 − dim Q + ∑(p_{i}^{a_{i}} − 1).

As remarked in §7, the conjecture is a theorem in the case that the index is 6 ([CTKM06, Theorem Theorem 1.3]). We therefore have that

ed(BG/k) ≥ 4 − dim Q .

Remark 9.4. Suppose G is a simple algebraic group whose center Z is cyclic. It is tempting to apply Theorem 9.2 to the natural sequence

1 ⟶ Z ⟶ G ⟶ G^{ad} ⟶ 1

where the adjoint group G^{ad} is G/Z. Given a torsor t ∈ H^{1}(K, G^{ad}), the central simple algebra representing ∂_{K}(t) ∈ H^{2}(K, Z) is called the Tits algebra of t. The possible values of the index of the Tits algebra were studied in [Tit92], where it is denoted by b(X) (for group of type X) and its possible values are listed on p. 1133. A quick look at this table reveals that for most types these indices are smaller than dim(G), so that the bound of Theorem 9.2 becomes vacuous. The only exception are groups of types B and D, in which case Theorem 9.2 does indeed, give interesting bounds; cf. Remark 13.7.

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

Our first application of Theorem 9.2 is to finish the proof of Theorem 1.8 from the introduction by showing that ed M_{1,0} = +∞.

Note that by M_{1,0} we mean the moduli stack of genus 1 curves, not the moduli stack M_{1,1} of elliptic curves (which is Deligne-Mumford). The objects of M_{1,0} are torsors for elliptic curves as opposed to the elliptic curves which appear as the objects of M_{1,1}. We will now see that these torsors are what causes the essential dimension to be infinite.

10.1. 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}.

Theorem 10.2. Let E = E_{q}/K denote the Tate curve over a field K as in (10.1). Then

ed E = +∞.

Theorem 10.2 is an immediate consequence of the following statement.

Lemma 10.3. Let E = E_{q} be a Tate curve as in (10.1) 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, Theorem 9.2 implies that ed BE[l^{n}] ≥ l^{n}.

To see that ed BE[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, by Theorem 3.9, we have the desired inequality. ♠

Proof of Theorem 10.2. For each prime power l^{n}, the morphism BE[l^{n}] → BE is representable of fiber dimension 1. We therefore have

ed E ≥ ed BE[l^{n}]
= l^{n} − 1

for all n. ♠

G. Pappas pointed out the following corollary.

Corollary 10.4. Let E be a curve over a number field K. Assume that there is at least one prime 𝔭 of K where E has semistable bad reduction. Then ed E = +∞.

It seems reasonable to make the following guess.

Conjecture 10.5. If E is an elliptic curve over a number field, then ed E = +∞.

Remark 10.6. Note, however, that, if A is a d-dimension complex abelian variety, then ed A = 2d; see [Bro].

Now we can complete the proof of Theorem 1.8.

Theorem 10.7. Let k be a field. Then ed(M_{1,0}/k) = +∞.

Proof. Consider the morphism M_{1,0} → M_{1,1} which sends a genus 1 curve to its Jacobian. Let F = k((t)) and 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:

(10.8)  \xymatrix{BE \ar[r] \ar[d] & M_{1,0} \ar[d] \\ Spec F \ar[r] & M_{1,1}.}

It follows that +∞ = ed BE ≤ ed M_{1,0}. ♠

11. Essential dimension of p-groups I

The goal of this section is to prove the following theorem from the introduction (Theorem 1.11).

Theorem 11.1. Let G be a p-group whose commutator [G, G] is central and cyclic. Then

(a) We have

ed_{k} G ≥ \sqrt{|G/C(G)|} + rank C(G) − 1

for any base field k of characteristic ≠ p.

(b) Moreover, assume that k contains a primitive root of unity of degree exp(G). Then G has a faithful representation of degree

\sqrt{|G/C(G)|} + rank C(G) − 1 .

Theorem 1.11 is an immediate consequence of this result, since part (b) implies ed_{k} G ≤ \sqrt{|G/C(G)|} + rank C(G) − 1; cf. e.g., Theorem 3.9. Our proof of part (a) will rely on the following lemma.

Lemma 11.2. Let G be a finite group and H be a central cyclic subgroup. Assume that there exists a character χ: G → k^{∗} whose restriction to H is faithful. Then

(1) ed(G) ≥ ed(G/H).

(2) Moreover, if H has prime order and is not properly contained in another central cyclic subgroup of G then ed(G) = ed(H) + 1.

Proof. Part (2) is proved in [BR97, Theorem 5.3] in characteristic zero and in [Kan06, Theorem 4.5] in prime characteristic.

To prove (1), let ϕ: G → G/H ↪ GL(V) be a faithful representation of G/H. Then ϕ⊕χ: G → GL(V ×k) is a faithful representation of G. Denote the class of the G-action on V × k by α ∈ H^{1}(K, G), where K = k(V ⊕ k)^{G}. Let β be the image of α in H^{1}(K, G/H). Then β is given by the induced action of G/H on

(V × k)/H ≃ V × k .

Here the quotient map V × k → V × k is given by (v, x) → (v, x^{d}), where d = |H|. This shows that induced action of G/H on (V × k)/H is again linear. Hence, β is a versal G/H-torsor; cf. [GMS03, Example 5.4] or [BR97, Theorem 3.1]. We conclude that

ed(G) = ed(α) ≥ ed(β) = ed(G/H) ,

as claimed. ♠

We now proceed with the proof of Theorem 11.1. We begin with the following reduction.

Lemma 11.3. In the course of proving Theorem 11.1 we may assume without loss of generality that the center C(G) is cyclic.

Proof. Let Z be a maximal cyclic subgroup of C(G) containing [G, G]. Then C(G) = Z ⊕ W for some central subgroup W of G. Note that rank(W) = rank C(Z) − 1. Moreover, W projects isomorphically into the abelian subgroup G/Z. In particular, if H is a cyclic subgroup of W then after composing this projection with a suitable character of G/Z, we obtain a character χ_{H}: G → k^{∗}, which is faithful on H. This means that Lemma 11.2 can be used to compare the essential dimensions of G and G/H.

We will now argue by induction on |W |. If |W | = 1, we are done. For the induction step we will choose a cyclic subgroup H ⊂ W (in a way, to be specified below) and assume that parts (a) and (b) of Theorem 11.1 hold for \overline{G} = G/H. Our goal will then be to prove that they also hold for G.

It is easy to see that g_{1} and g_{2} commute in G if and only if their images \overline{g_{1}} and \overline{g_{2}} commute in \overline{G}. In particular, C(\overline{G}) = C(G)/H ≃ Z ⊕ W/H. (Here Z projects isomorphically to a cyclic subgroup \overline{Z} of \overline{G} = G/H, and we are identifying Z with \overline{Z}). Consequently,

(11.4)  |\overline{G}/C(\overline{G})| = |G/C(G)| and rank C(\overline{G}) = rank(W/H) + 1.

(a) We choose H to be a subgroup of prime order in W. By induction assumption,

ed(\overline{G}) ≥ \sqrt{|\overline{G}/C(\overline{G})|} + rank(C(\overline{G})) − 1 .

We will now consider two cases.

Case 1. H is properly contained in another cyclic subgroup of W. In this case rank(W/H) = rank(W) and by (11.4)

rank C(\overline{G}) = rank(W/H) + 1 = rank(W) + 1 = rank C(G) .

By Lemma 11.2(1), ed(G) ≥ ed(\overline{G}) and thus

ed(G) ≥ \sqrt{|\overline{G}/C(\overline{G})|} + rank(C(\overline{G})) − 1
= \sqrt{|G|/|C(G)|} + rank C(G) − 1 ,

as desired.

Case 2. H is not properly contained in any cyclic subgroup of W. In this case rank(W/H) = rank(W) − 1 and by (11.4)

rank C(\overline{G}) = rank(W/H) + 1 = rank(W) = rank C(G) − 1 .

By Lemma 11.2(2), ed(G) = ed(\overline{G}) + 1 and thus

ed(G) = ed(\overline{G}) + 1
= \sqrt{|\overline{G}/C(\overline{G})|} + rank(C(\overline{G}))
= \sqrt{|G|/|C(G)|} + (rank(C(G)) − 1) .

(b) Here we choose H to be a maximal central cyclic subgroup of W (not necessarily of prime order). By our induction assumption, G/H has a representation ρ: G/H ↪ GL(V) of dimension

\sqrt{|\overline{G}/C(\overline{G})|} + rank(C(\overline{G})) − 1 .

Then ρ ⊕ χ_{H} is a faithful representation of G of dimension

\sqrt{|\overline{G}/C(\overline{G})|} + rank C(\overline{G}) = \sqrt{|G|/|C(G)|} + rank(C(G)) − 1 ;

cf. (11.4). This shows that Theorem 11.1 holds for G. ♠

Proof of Theorem 11.1(a). By Lemma 11.3 we may assume that C(G) is cyclic. Set Z = C(G). Since we are assuming that [G, G] ⊂ Z, the quotient A =^{def} G/Z is abelian.

In this case Theorem 1.11 reduces to ed(G) ≥ \sqrt{|A|}. By Theorem 9.2 it suffices to show that

ind(G, Z) ≥ \sqrt{|A|} .

so we will now direct our attention towards computing ind(G, Z).

We will use additive notation for the groups Z and A, multiplicative for G. In this situation we can define a skew-symmetric bilinear form ω: A×A → Z by

ω(a_{1}, a_{2}) = g_{1}g_{2}g_{1}^{−1}g_{2}^{−1} ,

where a_{i} = g_{i}, modulo Z, for i = 1, 2. (Note that ω(a_{1}, a_{2}) is independent of the choice of g_{1} and g_{2}.) Clearly g lies in C(G) if and only if its image a lies in the kernel of ω, i.e., ω(a, b) = 0 for every b ∈ A. Since we are assuming that C(G) = Z, we conclude that the kernel of ω is trivial, i.e., ω is a symplectic form on A. It is well known (see for example [TA86, §3.1]) that the order of A, which equals the order of G/C(G), is a complete square.

Fix a generator z of Z. We recall the basic result on the structure of a symplectic form ω on a finite abelian group A (the proof is easy; it can be found, e.g., in [Wal63, §3.1] or [TA86, §7.1]). There exist elements a_{1}, …, a_{2r} in A and positive integers d_{1}, …, d_{r} with the following properties.

(a) d_{i} divides d_{i−1} for each i = 2, …, r, and d_{r} > 1.

(b) Let i be an integer between 1 and r. If A_{i} denotes the subgroup of A generated by a_{i} and a_{r+i}, then there exists an isomorphism \overline{A}_{i} ≃ (ℤ/d_{i}ℤ)^{2} such that a_{i} corresponds to (1,0) and a_{r+i} to (0, 1).

(c) The subgroups A_{i} are pairwise orthogonal with respect to ω.

(d) ω(a_{i}, a_{r+i}) = z^{n/d_{i}} ∈ Z.

(e) A = A_{1} ⊕ ⋯ ⊕ A_{r}.

Then the order of A is d_{1}^{2} … d_{r}^{2}, hence \sqrt{|A|} = d_{1} … d_{r}.

Let G_{i} be the inverse image of A_{i} in G; note that G_{i} commutes with G_{j} for any i ≠ j.

Let u_{1}, …, u_{2r} be indeterminates, and set K =^{def} k(u_{1}, …, u_{2r}). Identify Z with μ_{n} by sending z into ζ_{n}. Consider the boundary map

∂_{i}: H^{1}(K, A_{i}) ⟶ H^{2}(K, Z)

obtained from the exact sequence

1 ⟶ Z ⟶ G_{i} ⟶ A_{i} ⟶ 1.

Claim. There exists a class ξ_{i} ∈ H^{1}(K, A_{i}) such that ∂_{i}ξ_{i} is the class of the cyclic algebra (u_{i}, u_{r+i})_{d_{i}} in Br K.

To see that Theorem 11.1(a) follows from the claim, consider the commutative diagram

\xymatrix{1 \ar[r] & Z^{r} \ar[r] \ar[d]^{m} & ∏_{i} G_{i} \ar[r] \ar[d] & ∏_{i} A_{i} \ar[r] \ar[d] & 1 \\ 1 \ar[r] & Z \ar[r] & G \ar[r] & A \ar[r] & 1}

in which m is defined by the formula m(z_{1}, …, z_{r}) = z_{1} … z_{r}, and the homomorphism ∏_{i} G_{i} → G is induced by the inclusions G_{i} ⊆ G. This yields a commutative diagram

\xymatrix{∏_{i} H^{1}(K, A_{i}) \ar[r]^{∏_{i} ∂_{i}} \ar[d] \ar[r] & H^{2}(K, Z)^{r} \ar[d]^{m_{∗}} \\ H^{1}(K, A) \ar[r]^{∂} & H^{2}(K, Z)}

in which the map m_{∗} is such that m(α_{1}, …, α_{r}) = α_{1} … α_{r}. So, if ξ ∈ H^{1}(K, A) is the image of (ξ_{1}, …, ξ_{r}), we have that ∂ξ is the class of the product

(u_{1}, u_{r+1})_{d_{1}} ⊗_{K} (u_{2}, u_{r+2})_{d_{2}} ⊗_{K} ⋯ ⊗_{K} (u_{r}, u_{2r})_{d_{r}},

whose index is d_{1} … d_{r}. Hence ind(G, Z) ≥ d_{1} … d_{r} = \sqrt{|A/K|}, as needed.

Now we prove the claim. Choose a power of p, call it d, that is divisible by the order of Z and by the order of each a_{i}. Consider the group Λ(d) defined by the presentation

⟨x_{1}, x_{2}, y | x_{1}^{d} = x_{2}^{d} = y^{d} = 1, x_{1}x_{2} = yx_{2}x_{1}, x_{1}y = yx_{1}, x_{2}y = yx_{2}⟩.

Call ρ_{i}: Λ(d) → G_{i} the homomorphism obtained by sending x_{1} to a_{i}, x_{2} to a_{r+i}, and y to z^{n/d_{i}} = ω(a_{i}, a_{r+i}).

Let ζ_{d} be a primitive d-th root of 1 in k such that ζ_{n} = ζ_{d}^{n/d}. The subgroup ⟨y⟩ in Λ(d) is cyclic of order d; we fix the isomorphism ⟨y⟩ ≃ μ_{d} so that y corresponds to ζ_{d}. The restriction of ρ_{i} to ⟨y⟩ → Z corresponds to the homomorphism μ_{d} → μ_{n} defined by α ↦ α^{d/d_{i}}. We have a commutative diagram

\xymatrix{1 \ar[r] & μ_{d} \ar[r] \ar[d]_{\begin{matrix}α \\ ↓ \\ α^{d/d_{i}} \end{matrix}} & Λ(d) \ar[r] \ar[d]^{ρ_{i}} & (ℤ/dℤ)^{2} \ar[d] \ar[r] & 1 \\ 1 \ar[r] & μ_{n} \ar[r] & G_{i} \ar[r] & A_{i} \ar[r] & 1.}

We have H^{1}(K, (ℤ/dℤ)^{2}) = (K^{∗}/K^{∗}^{d})^{2}. According to [Vel00, Example 7.2], the image of the element (u_{i}, u_{r+i}) ∈ H^{1}(K, (ℤ/dℤ)^{2}) is the cyclic algebra (u_{i}, u_{r+i})_{d}; hence, if ξ_{i} is the image in H^{1}(K, A_{i}) of (u_{i}, u_{r+i}), the image of ξ_{i} in H^{2}(K, μ_{d}) is the algebra (u_{i}, u_{r+i})_{d}^{⊗d/d_{i}}, which is equivalent to (u_{i}, u_{r+i})_{d_{i}}. This concludes the proof of Theorem 11.1(a). ♠

Proof of Theorem 11.1(b). By Lemma 11.3 we may assume that C(G) = Z is cyclic. In this case Theorem 11.1 asserts that G has a faithful representation of degree \sqrt{|A|}.

Suppose |C(G)| = p^{h} and |A| = p^{2m}; we want to construct a faithful representation of G of dimension p^{m}. By [TA86, §3.1] A contains a Lagrangian subgroup L of order p^{m}. Denote by H the inverse image of L in G; then H is an abelian subgroup of G of order p^{h+m}. Since ζ_{p^{e}} ∈ k we can embed Z in k^{∗} and extend this embedding to a homomorphism χ: H → k^{∗}. We claim that the representation ρ: G → GL_{p^{m}} induced by χ is faithful.

It is enough to show that ρ(g) ≠ id for any g ∈ G of order p, or, equivalently, that ρ |_{⟨g⟩} is non-trivial for any such g. If s ∈ G consider the subgroup H_{s} =^{def} s⟨g⟩s^{−1} ∩ H of H, which is embedded in ⟨g⟩ via the homomorphism x ↦ s^{−1}xs. By Mackey’s formula ([Ser77, §7.3]), ρ |_{⟨g⟩} contains all the representations of ⟨g⟩ induced by the restrictions χ |_{H_{s}} via the embedding above.

If g ∉ H then H_{1} = ⟨g⟩ ∩ H = {1}: we take s = 1, and we see that ρ |_{⟨g⟩} contains a copy of the regular representation of ⟨g⟩, which is obviously non-trivial.

Assume g ∈ H. Then H_{s} = ⟨sgs^{−1}⟩ for any s ∈ G; it is enough to prove that χ(sgs^{−1}) ≠ 1 for some s ∈ G. If χ(g) ≠ 1 then we take s = 1. Otherwise χ(g) = 1; in this case g ∉ C(G), because χ |_{C(G)} : C(G) → k^{∗} is injective. Hence the image \overline{g} of g in A is different from 0, and we can find s ∈ G such that ω(\overline{s}, \overline{g}) ≠ 1. Then we have

χ(sgs^{−1}) = χ(ω(\overline{s}, \overline{g})g) = χ(ω(\overline{s}, \overline{g}))χ(g) = χ(ω(\overline{s}, \overline{g})) ≠ 1.

This concludes the of Theorem 1.11(b). ♠

12. Essential dimension of p-groups II

We begin with several simple illustrations of Theorem 1.11.

Example 12.1. Recall that a p-group G is called extra-special if its center Z is cyclic of order p, and the quotient G/Z is elementary abelian. The order of an extra special p-group G is an odd power of p; the exponent of G is either p or p^{2}; cf. [Rob96, pp. 145–146]. Note that every non-abelian group of order p^{3} is extra-special. For extra-special p-groups Theorem 1.11 reduces to the following.

Let G be an extra-special p-group of order p^{2m+1}. Assume that that the characteristic of k is different from p, that ζ_{p} ∈ k, and ζ_{p^{2}} ∈ k if the exponent of G is p^{2}. Then ed G = p^{m}.

Example 12.2. Let p be an odd prime and G = C_{p^{r}} ⋉ C_{p^{s}} be the natural semidirect product of cyclic groups of order p^{r} and p^{s} (in other words, C_{p^{s}} is identified with the unique subgroup of C_{p^{r}}^{∗} of order p^{s}). If s ≤ r/2 then

ed_{k}(C_{p^{r}} ⋉ C_{p^{s}}) = p^{s} ,

for any field k containing a primitive pth root of unity ζ_{p}.

Proof. Here C(G) is the (unique) subgroup of C_{p^{r}} of order p^{s}. If s ≤ r/2, this subgroup is central. Thus, if ζ_{p^{r}}, the equality ed_{k}(G) = p^{s} is an immediate consequence of Theorem 1.11. Since we are only assuming that ζ_{p} ∈ k, Theorem 1.11 only tells us that ed_{k}(G) ≥ p^{s}. To prove the opposite inequality, we argue as follows. Let F be the prime subfield of k. By [Led02, Corollary to Proposition 2], ed_{F (ζ_{p})}(G) ≤ p^{s}. Since we are assuming that F (ζ_{p}) ⊂ k, we conclude that ed_{k}(G) ≤ p^{s} as well. ♠

Corollary 12.3. Suppose k is a base field of characteristic ≠ p. If G is a non-abelian finite p-group then ed G ≥ p.

Proof. We argue by contradiction. Assume the contrary and let G be a nonabelian p-group G of smallest possible order such that ed G < p. Since G has a non-trivial center, there exists a cyclic central subgroup Z ⊂ G. The short exact sequence

(12.4)  1 ⟶ Z ⟶ G ⟶ G/Z ⟶ 1

give rise to the exact sequence of pointed sets

H^{1}(K, G) ⟶ H^{1}(K, G/Z) ⟶^{∂_{K}} H^{2}(K, Z)

for any field extension K of our base field k. We will now consider two cases.

Case 1. Suppose the map H^{1}(K, G) → H^{1}(K, G/Z) is not surjective for some K/k. Then ∂_{K} is non-trivial, and Theorem 9.2 tells us that ed G ≥ p, a contradiction.

Case 2. Suppose the map H^{1}(K, G) → H^{1}(K, G/Z) is surjective for every K/k. Then the morphism BG → B(G/Z) is isotropic, and Proposition 2.22 implies that p > ed G ≥ ed(G/Z). By the minimality of G, the group G/Z has to be abelian. Consequently, [G, G] ⊂ Z is cyclic and central in G. Since G is non-abelian, |G/C(G)| ≥ p^{2}. Theorem 1.11 now tells us that

ed(G) = \sqrt{|G|/|C(G)|} + rank C(G) − 1 ≥ p ,

a contradiction. ♠

We will conclude this section by answering the following question of Jensen, Ledet and Yui [JLY02, p. 204].

Question 12.5. Let G be a finite group and N be a normal subgroup. Is it true that ed G ≥ ed(G/N)?

The inequality ed(G) ≥ ed(G/N) is known to hold in many cases (cf., e.g., Lemma 11.2). We will now show that it does not hold in general, even if H is assumed to be central.

Corollary 12.6. For every real number λ > 0 there exists a finite p-group G, with a central subgroup H ⊂ G such that ed(G/H) > λ ed G.

Proof. Let Γ be a non-abelian group of order p^{3}. The center of Γ has order p; denote it by C. The center of Γ^{n} = Γ × ⋯ × Γ (n times) is then C^{n}. Let H_{n} be the subgroup of C^{n} consisting of n-tuples (c_{1}, …, c_{n}) such that c_{1} … c_{n} = 1. Clearly

ed Γ^{n} ≤ n · ed Γ = np;

see Example 12.1.

On the other hand, G_{n}/H_{n}, is easily seen to be extra-special of order p^{2n+1}, so ed(G_{n}/H_{n}) = p^{n}, again by Example 12.1. Hence by taking n sufficiently large we finish the proof. ♠

13. Spinor groups

In this section we will prove Theorem 1.13 stated in the introduction. For an introduction to the structure of this group and the theory of Clifford algebras and spin modules, we refer the reader to [FH91, §20.2] and [Che54].

Recall that by Spin_{n} we mean the totally split form of the spin group in dimension n over a field of characteristic not equal to 2. Let us be explicitly about this.

Write ⟨a_{1}, …, a_{n}⟩ for the rank n-quadratic form q given by q(x_{1}, …, x_{n}) = ∑_{i=1}^{n} a_{i}x_{i}^{2}. Set h to be the standard hyperbolic quadratic form given by h(x, y) = xy. (Thus h ≅ ⟨1, −1⟩). For each n ≥ 0 define

(13.1)  h_{n} = \begin{cases}h_{n}^{⊕n/2}, & if n is even, \\ h_{n}^{⊕(n−1/2)} ⊕ ⟨1⟩, & if n is odd. \end{cases}

Set O_{n} =^{def} O(h_{n}), SO_{n} =^{def} SO(h_{n}), and Spin_{n} = Spin(h_{n}). These are all totally split groups.

Now, one of the hypotheses of Theorem 1.13 is that ζ_{4} ∈ k. Therefore we can write Spin_{n} as Spin(q), where

q(x_{1}, …, x_{n}) = −(x_{1}^{2} + ⋯ + x_{n}^{2}).

Consider the subgroup Γ_{n} ⊆ SO_{n} consisting of diagonal matrices, which is isomorphic to μ_{2}^{n−1}. Call G_{n} the inverse image of Γ_{n} in Spin_{n}. It is a constant group scheme over k. Denote by μ_{2} the kernel of the homomorphism Spin_{n} → SO_{n}.

Lemma 13.2. Every Spin_{n}-torsor over an extension K of k admits a reduction of structure group to G_{n}.

Proof. Let P → Spec K be a Spin_{n}-torsor: we are claiming that the Kscheme P/G_{n} has a rational point. We have P/G_{n} = (P/μ_{2})/Γ_{n}. However P/μ_{2} → Spec K is the SO_{n} torsor associated with P → Spec K, and every SO_{n}-torsor has a reduction of structure group to Γ_{n}. ♠

This means that the natural morphism BG_{n} → BSpin_{n} is isotropic; so from Propositions 3.3 and 2.22 we get the bounds

(13.3)  ed G_{n} − dim Spin_{n} ≤ ed Spin_{n} ≤ ed G_{n}.

Of course dim Spin_{n} = n(n−1)/2; we need to compute ed G_{n}. The structure of G_{n} is well understood (besides the references cited above, it is also very clearly described in [Woo89]). The group scheme Spin_{n} is a subgroup scheme of the group scheme of units in the Clifford algebra A_{n} of the quadratic form −(x_{1}^{2} + ⋯ + x_{n}^{2}). The algebra A_{n} is generated by elements e_{1}, …, e_{n}, with relations e_{i}^{2} = −1 and e_{i}e_{j} + e_{j}e_{i} = 0 for all i ≠ j. The element e_{i} is in Pin_{n}, and image of e_{i} in O_{n} is the diagonal matrix with −1 as the i-th diagonal entry, and 1 as all the other diagonal entries. The kernel of the homomorphism Pin_{n} → O_{n} is {±1}.

For any I ⊆ {1, …, n} write I = {i_{1}, …, i_{r}} with i_{1} < i_{2} < ⋯ < i_{r} and set e_{I} =^{def} e_{i_{1}} … e_{i_{r}}. The group G_{n} consists of the elements of A_{n} of the form ±e_{I}, where I ⊆ {1, …, n} has an even number of elements. The element −1 is central, and the commutator [e_{I}, e_{J}] is given by

[e_{I}, e_{J}] = (−1)^{|I∩J|}

It is clear from this description that G_{n} is a 2-group, of order 2^{n}, the commutator [G_{n}, G_{n}] = {±1} is cyclic, and the center C(G) is given by

C(G_{n}) = \begin{cases}\{±1\} ≃ ℤ/2ℤ, if n is odd, \\ \{±1, ±e_{\{1,...,n\}}\} ≃ ℤ/4ℤ, if n ≡ 2 (mod 4), \\ \{±1, ±e_{\{1,...,n\}}\} ≃ ℤ/2ℤ × ℤ/2ℤ, if n is divisible by 4. \end{cases}

Theorem 1.11 now tells us that

ed(G_{n}) = \begin{cases}2^{(n−1)/2}, if n is odd, \\ 2^{(n−2)/2}, if n ≡ 2 (mod 4), \\ 2^{(n−2)/2} + 1, if n is divisible by 4. \end{cases}

Substituting this into (13.3), we obtain the bounds of Theorem 1.13. ♠

Remark 13.4. The same argument, with G_{n} replaced by the inverse image of the diagonal subgroup of O_{n} in Pin_{n}, yields the following bounds on the essential dimensions of Pin groups (over a field k satisfying the assumptions of Theorem 1.13):

2^{⌊n/2⌋} − \frac{n(n − 1)}{2} ≤ ed Pin_{n} ≤ 2^{⌊n/2⌋}, if n ≢ 1 (mod 4),
2^{⌊n/2⌋} − \frac{n(n − 1)}{2} + 1 ≤ ed Pin_{n} ≤ 2^{⌊n/2⌋} + 1, if n ≡ 1 (mod 4).

Remark 13.5. When n ≤ 14 the lower bound of Theorem 1.13 is negative and the upper bound is much larger than the true value of ed Spin_{n}. For n = 15 and 16 our inequalities yield

23 ≤ ed Spin_{15} ≤ 128

and

9 ≤ ed Spin_{16} ≤ 129.

When n = 16 our lower bound coincides with the lower bound (1.14) of Reichstein–Youssin and Chernousov–Serre, while for n = 15 it is substantially larger. When n ≥ 17 the exponential part of the lower bound takes over, the growth becomes fast and the gap between the lower bound and the upper bound proportionally small. For values of n close to 15 our estimates are quite imprecise; it would be interesting to improve them.

Remark 13.6. By Proposition 2.12, the lower bounds in the theorem hold for over any field of characteristic different from 2 (and for any form of the Spin group).

On the other hand, if we do not assume that ζ_{4} ∈ k, we get the slightly weaker upper bound

ed Spin_{n} ≤ 2^{⌊(n−1)/2⌋} + n − 1

for the totally split form of the spin group in dimension n. To prove this inequality, we observe that a generically free representation of Spin_{n} can be constructed by taking a spin, or half-spin, representation V of Spin_{n} of dimension 2^{⌊(n−1)/2⌋}, and adding a generically free representation W of SO_{n}. Since the essential dimension of SO_{n−1} is n − 1 over any field of characteristic different from 2, there is an SO_{n}-compression f: W −−→ X, where dim(X) = dim(SO_{n}) + n − 1. Now id × f: V × W −−→ V × X is a Spin_{n}-compression of V × W. Consequently,

ed Spin_{n} ≥ dim(V × X) − dim Spin_{n}
= 2^{⌊(n−1)/2⌋} + dim SO_{n} + n − 1 − dim Spin_{n}
= 2^{⌊(n−1)/2⌋} + n − 1 ,

as claimed.

Remark 13.7. It is natural to ask whether the inequality

ed Spin_{n} ≥ 2^{⌊(n−1)/2⌋} − \frac{n(n − 1)}{2}

can be proved by a direct application of Theorem 1.13 to the exact sequence

(13.8)  1 ⟶ μ_{2} ⟶ Spin_{n} ⟶ SO_{n} ⟶ 1

without considering the finite subgroup G_{n} of Spin_{n}. The answer is “yes.” Note, however, that we see no way of obtaining the slightly stronger lowerbound which we obtain when n is divisible by 4.

Indeed, consider the associated coboundary map

\xymatrix{H^{1}(K, SO_{m}) \ar[r]^{∂_{K}} & H^{2}(K, μ_{2}).}

A class in H^{1}(K, SO_{m}) is represented by a m-dimensional quadratic form q of discriminant 1 defined over K. The class of ∂_{K}(q) ∈ H^{2}(K, μ_{2}) is then the Hasse-Witt invariant of q; following Lam [Lam73], we will denote it by c(q). (Note that since we are assuming that −1 is a square in k, the Hasse invariant and the Witt invariant coincide; see [Lam73, Proposition V.3.20].) Our goal is thus to show that for every n ≥ 1 there exists a quadratic form q_{n} of dimension n and discriminant 1 such that c(q_{n}) has index 2^{⌊(n−1)/2⌋}.

If n is even this is proved in [Mer91, Lemma 5]. (Note that in this case c(q) ∈ H^{2}(K, μ_{2}) is the class of the Clifford algebra of q.) If n = 2m + 1 is odd, set K = k(a_{1}, b_{1}, …, a_{m}, b_{m}), where a_{1}, b_{1}, …, a_{m}, b_{m} are independent variables, and define q_{n} recursively by

q_{3} = ⟨a_{1}, b_{1}, a_{1}b_{1}⟩ and q_{n+2} = ⟨a_{n}b_{n}⟩ ⊗ q_{n} ⊕ ⟨a_{n}, b_{n}⟩.

A direct computation using basic properties of the Hasse-Witt invariant (see, e.g., [Lam73, Section V.3]) shows that c(q_{2m+1}) is the class of the product (a_{1}, b_{1})_{2} ⊗_{K} ⋯ ⊗_{K} (a_{m}, b_{m})_{2} of quaternion algebras. This class has index 2^{m}, as claimed. ♠

Any lower bound on ed Spin_{n} obtained in this way (i.e., directly from Theorem 9.2) will necessarily be of the form ed Spin_{n} ≥ 2^{m} − \frac{n(n−1)}{2} for some integer m ≥ 0. Thus, while this approach recovers the lower bound of Theorem 1.13 if n is not divisible by 4, it cannot be used to do so if n is divisible by 4.

To conclude this section, we will now prove similar bounds on the essential dimensions on half-spin groups. We begin with the following simple corollary of [CGR06, Theorem 1.1], which appears to have been previously overlooked.

Lemma 13.9. Let G be a closed (but not necessarily connected) subgroup of GL_{n} defined over a field k. Assume that one of the following conditions holds.

(1) k is algebraically closed of characteristic 0.

(2) k has characteristic 0 and G is connected.

(3) G is connected and reductive.

Then ed G ≤ n.

Proof. According to [CGR06, Theorem 1.1], there exists a finite subgroup scheme S ⊆ G such that every G-torsor over Spec K, where K is an extension of G, has a reduction of structure groups to S. Hence the morphism BS → BG is isotropic, so ed G ≤ ed S. But the restriction of the representation G ⊆ GL_{n} is generically free, hence ed S ≤ n. ♠

Example 13.10. Suppose G satisfies one of the conditions (a), (b) or (c) of Lemma 13.9 and the centralizer C_{G}(G^{0}) of the connected component of G is trivial. Then the adjoint representation of G is faithful and Lemma 13.9 tells us that ed(G) ≤ dim(G). In particular, this inequality is valid for every connected semisimple adjoint group G. (In the case of simple adjoint groups, a stronger bound is given by [Lem04, Theorem 1.3].)

We are now ready to proceed with our bounds on the essential dimension of half-spin groups. Recall that the half-spin group HSpin_{n} is defined, for every n divisible by 4, as Spin_{n}/⟨η⟩, where η is an element of the center of Spin_{n} different from −1. (There are two such elements, but the resulting quotients are isomorphic.)

Theorem 13.11. Suppose k is a field of characteristic ≠ 2 and ζ_{4} ∈ k. Let n be a positive integer divisible by 4. Then

2^{(n−2)/2} − \frac{n(n − 1)}{2} ≤ ed HSpin_{n} ≤ 2^{(n−2)/2}

Proof. The group HSpin_{n} contains G_{n}/⟨η⟩ ≃ G_{n−1}, which is an extra-special group of order 2^{n−1}. By Example 12.1 ed(G_{n}/⟨η⟩) = 2^{(n−2)/2} and thus

ed HSpin_{n} ≥ ed(G_{n}/⟨η⟩) − dim HSpin_{n}

= 2^{(n−2)/2} − \frac{n(n − 1)}{2} ,

as in the proof of Theorem 1.13.

For the upper bound notice that one of the two half-spin representations of Spin_{n} descends to HSpin_{n}, and is a faithful representation of HSpin_{n} of dimension 2^{(n−2)/2}. The upper bound now follows from Lemma 13.9 ♠

14. Essential dimension of cyclic p-groups

In this section, we are going to prove the following theorem due to M. Florence. In the sequel p will denote a prime, different from the characteristic of our base field k, and ζ_{d} will denote a primitive dth root of unity in \overline{k}. Recall that we have set C_{n} =^{def} ℤ/nℤ.

Theorem 14.1 (M. Florence [Flo06]). Let p be a prime, k a field of characteristic ≠ p. Suppose ζ_{p^{n}} ∈ k but ζ_{p^{n+1}} ∉ k for some integer n ≥ 1. Moreover, if p = 2 and n = 1, assume also that k(ζ_{4}) ≠ k(ζ_{8}). Then

ed C_{p^{m}} = \begin{cases}p^{m−n} & if n < m, \\ 1 & if n ≥ m. \end{cases}

for any integer m ≥ 1,

This theorem was independently obtained by us in the case where n ≥ ⌊(m + 1)/2⌋; in particular, for m = 2. However, our proof of the stronger result given by Theorem 14.1, will rely on an idea of M. Florence, in combination with the lower bound of Theorem 9.2.

Proof. If m ≤ n, then C_{p^{m}} = μ_{p^{m}}. Therefore ed C_{p^{m}} = 1 by [BF03, Example 2.3]. We can therefore restrict our attention to the case n < m.

We first show that ed C_{p^{m}} ≤ p^{m−n}. To do this, pick a faithful character χ: C_{p^{n}} → 𝔾_{m} defined over K and set V =^{def} Ind_{C_{p^{n}}}^{C_{p^{m}}} χ. A simple calculation shows that V is faithful, thus, V is generically free since C_{p^{m}} is finite. By Theorem 3.9, it follows that ed C_{p^{m}} ≤ dim V = p^{m−n}.

To show that ed C_{p^{m}} ≥ p^{m−n}, view the representation V as a homomorphism ρ: C_{p^{m}} → GL(V) of algebraic groups. Let π : GL(V) → PGL(V) denote the obvious projection and note that the kernel of π ∘ ρ is exactly C_{p^{n}}. It follows that we have a commutative diagram

\xymatrix{0 \ar[r] & C_{p^{n}} \ar[r] \ar[d] & C_{p^{m}} \ar[r]^{ρ} \ar[d] & C_{p^{m−n}} \ar[r] \ar[d]^{ι} & 1 \\ 1 \ar[r] & 𝔾_{m} \ar[r] & GL(V) \ar[r]^{π} & PGL(V) \ar[r] & 1}

where the rows are exact and the columns are injective.

Let K/k be a field extension and let t ∈ H^{1}(K, C_{p^{m}}) be a torsor. Let ι_{∗}: H^{1}(K, C_{p^{m−n}}) → H^{1}(K, PGL(V)) denote the map induced by ι. Then, from the commutativity of the above diagram (and the injectivity of the columns), it follows that ind_{K}(t) is the index of of the CSA ι_{∗}(t).

We claim that there is a t such that ι_{∗}(t) is a division algebra. From this it will clearly follow that ind_{K}(t) = dim V = p^{m−n}.

In fact, this t ∈ H^{1}(K, C_{p^{m}}) is simply the “generic” one. (This is the part of the argument that we learned from Florence’s preprint [Flo06].) Namely, let L = K(x_{1}, …, x_{p^{m−n}}) denote the field obtained by adjoining p^{m−n} independent variables to K, and let C_{p^{m−n}} act on L by permuting the variables in the obvious way (k · x_{i} = x_{i+k} (mod p^{m−n})). Let F = L^{C_{p^{m−n}}}. Then L/K defines a C_{p^{m−n}}-torsor t over K.

In the case where k = ℚ(ζ_{p^{n}}) the torsor ι_{∗}(t) is the “generic” algebra R_{p^{n},p^{m},p^{m}} of [Row88, §7.3]. By a theorem of Brauer (see Theorem [Row88, Theorem 7.3.8]) it is a division algebra. A similar argument (due to M. Florence) shows that the same is true if ℚ(ζ_{p^{n}}) is replaced by our field k (satisfying the assumptions of Theorem 14.1). ♠

Remark 14.2. Suppose ζ_{6} ∈ k. Note that in this case [k(ζ_{36} : k] always divides 6. Assume [k(ζ_{36}) : k] = 6. (This occurs, for example, if k = ℚ(ζ_{6}).) We claim that ed_{k} C_{36} ≥ 4.

Indeed, by Remark 9.3 it suffices to show that ind(C_{36}, C_{6}) = 6. Let K/k be a field extension and consider the boundary map

∂_{K}: K^{∗}/(K^{∗})^{6} = H^{1}(K, C_{6}) ⟶ H^{2}(K, C_{6})

induced by the exact sequence 1 → C_{6} → C_{36} → C_{6} → 1. By [Vel00, Theorem 7.1], ∂_{K} sends (a) to the class of the cyclic algebra (a, ζ_{6})_{6}. The index of this cyclic algebra clearly divides 6. Taking K = k(a), where a is an independent variable over k, and applying Wedderburn’s criterion (cf. e.g., [Pie82, Corollary 15.1d]), we conclude that in this case the cyclic algebra (a, ζ_{6})_{6} has index 6. Thus ind(C_{36}, C_{6}) = 6, as claimed.

A similar argument shows that if Conjecture 7.4 is valid for n = p_{1}^{a_{1}} … p_{r}^{a_{r}} then ed_{k}(C_{n}) ≥ p_{1}^{a_{1}} + ⋯ + p_{r}^{a_{r}} − r + 1.

Let D_{n} be the dihedral group of order 2n. Ledet [Led02, Section 3] conjectured that if n is odd then ed C_{n} = ed D_{n} over any field k of characteristic zero. We will now prove this conjecture in the case where n = p^{r} is a prime power and k contains a primitive pth root of unity.

Corollary 14.3. Let p be an odd prime and k be a field containing a primitive pth root of unity. Then ed_{k} D_{p^{m}} = ed_{k} C_{p^{m}}.

Proof. If ζ_{p^{m}} ∈ k then we know that ed_{k} C_{p^{m}} = ed_{k} D_{p^{m}} = 1; see the proof of [BR97, Theorem 6.2]. Thus we may assume ζ_{p^{m}} ∉ k.

Let s be the largest integer n such that ζ_{p^{n}} ∈ k. By our assumption 1 ≤ s ≤ m − 1. By Theorem 14.1

ed_{k} C_{p^{m}} = p^{m−n} .

Since C_{p^{m}} ⊂ D_{p^{m}}, we clearly have ed_{k} D_{p^{m}} ≥ ed_{k} C_{p^{m}}. Thus we only need to show that

ed_{k} D_{p^{m}} ≤ p^{m−n} .

To prove this inequality, note that D_{p^{m}} ≃ C_{p^{m}} ⋊ C_{2} has a subgroup isomorphic to D_{p^{n}} = C_{p^{n}} ⋊ C_{2} of index p^{m−n}. Since k contains ζ_{p^{n}}, D_{p^{n}} has essential dimension 1 over k. Thus, by [Led02, Section 3],

ed_{k} D_{p^{m}} = ed_{k} D_{p^{n}} ≤ [D_{p^{m}} : D_{p^{n}}] = 1 · p^{m−n} = p^{m−n}.

This completes the proof of Corollary 14.3. ♠

15. Pfister numbers

Let k be a field of characteristic not equal to 2 and write W(k) for the Witt ring of k; see [Lam73, Chapter 2]. Let I = I(k) denote the ideal of all even dimensional forms in the Witt ring. Then, for any integer a > 0, I^{a} is generated as an abelian group by the a-fold Pfister forms [Lam73, Proposition 1.2].

Let q be a quadratic form of rank n > 0 whose class [q] in W(k) lies in I^{a} for a > 0. Define the a-Pfister number of q to be the minimum number r appearing in a representation

q = ∑_{i=1}^{r} ±p_{i}

with the p_{i} being a-fold Pfister forms. The (a, n)-Pfister number Pf_{k}(a, n) is the supremum of the a-Pfister number of q taken over all field extensions K/k and all n-dimensional forms q such that [q] ∈ I^{a}(K).

We have the following easy (and probably well-known) result.

Proposition 15.1. Let k be a field of characteristic not equal to 2 and let n be a positive even integer.

(a) Pf_{k}(1, n) ≤ n.

(b) Pf_{k}(2, n) ≤ n − 2.

Proof. (a) If n is even ⟨a_{1}, …, a_{n}⟩ = ∑_{i=1}^{n}(−1)^{i}≪−a_{i} ≫.

(b) Let q = ⟨a_{1}, …, a_{n}⟩ be an n-dimensional quadratic form over K. Recall that q ∈ I^{2}(K) iff n is even and d_{±}(q) = 1, modulo (K^{∗})^{2} [Lam73, Corollary II.2.2]. Here d_{±}(q) is the signed determinant given by (−1)^{n(n−1)/2}d(q) where d(q) = ∏_{i=1}^{n} a_{n} is the determinant [Lam73, p.38].

To explain how to write q as a sum of n − 2 Pfister forms, we will temporarily assume that ζ_{4} ∈ K. In this case we may assume that a_{1} … a_{n} = 1. Since ⟨a, a⟩ is hyperbolic for every a ∈ K^{∗}, we see that q = ⟨a_{1}, …, a_{n}⟩ is Witt equivalent to

≪ a_{2}, a_{1} ≫ ⊕ ≪ a_{3}, a_{1}a_{2} ≫ ⊕ ⋯ ⊕ ≪ a_{n−1}, a_{1} … a_{n−2} ≫ .

By inserting appropriate powers of −1, we can modify this formula so that it remains valid even if we do not assume that ζ_{4} ∈ K, as follows:

q = ⟨a_{1}, …, a_{n}⟩ ≃ ∑_{i=2}^{n}(−1)^{i}≪(−1)^{i+1}a_{i}, (−1)^{i(i−1)/2+1}a_{1} … a_{i−1} ≫ ♠

We do not have an explicit upper bound on Pf_{k}(3, n); however, we do know that Pf_{k}(3, n) is finite for any k and any n.

To explain this, let us recall that I^{3}(K) is the set of all classes [q] ∈ W(K) such that q has even dimension, trivial signed determinant and trivial Hasse-Witt invariant [KMRT98].

Let n be a positive integer. Let q be a non-degenerate n-dimensional quadratic form over K whose whose signed determinant is 1. The class of q in H^{1}(K, O_{n}) lies in H^{1}(K, SO_{n}). We say that q admits a spin structure if its class is in the image of H^{1}(K, Spin_{n}) into H^{1}(K, SO_{n}). As pointed out in Remark 13.7, the obstruction to admitting a spin structure is the Hasse-Witt invariant c(q). Thus, the forms in I^{3} are exactly the even dimensional forms admitting a spin structure. The following result was suggested to us by Merkurjev and Totaro.

Proposition 15.2. Let k be a field of characteristic different from 2. Then Pf_{k}(3, n) is finite.

Sketch of proof. Let E be a versal torsor for Spin_{n} over a field extension L/k; cf. [GMS03, Section I.V]. Let q_{L} be the quadratic form over L corresponding to E under the map H^{1}(L, Spin_{n}) → H^{1}(L, O_{n}). The (3, n)-Pfister number of q_{L} is then an upper bound for the (3, n)-Pfister number of any form over any field extension K/k. ♠

Remark 15.3. For a > 3 the finiteness of Pf_{k}(a, n) is an open problem.

The main theorem in this section is a lower bound for Pf_{k}(3, n) stated as Theorem 1.15 in the Introduction. We restate it here for the reader’s convenience.

Theorem 15.4. Let k be a field of characteristic different from 2 and let n be an even positive integer. Then

Pf_{k}(3, n) ≥ \frac{2^{(n+4)/4} − n − 2}{7} .

For each extension K of k, denote by T_{n}(K) the image of H^{1}(K, Spin_{n}) into H^{1}(K, SO_{n}); we get a functor T_{n}: Fields_{k} → Sets. The essential dimension of this functor is closely related to the essential dimension of Spin_{n}.

Lemma 15.5. ed Spin_{n} − 1 ≤ ed T_{n} ≤ ed Spin_{n}.

Proof. In the language off [BF03, Definition 1.12], we have a fibration of functors

H^{1}(−, μ_{2}) ⇝ H^{1}(−, Spin_{n}) ⟶ T_{n}(K).

The first inequality then follows from [BF03, Proposition 1.13] and the second follows from Proposition 2.22. ♠

Lemma 15.6. Let q and q′ be non-degenerate quadratic forms over K. Suppose that q admits a spin structure. Then q ⊕ q′ admits a spin structure if and only if q′ admits a spin structure.

Proof. Immediate from the identity c(q ⊕ q′) = c(q) + c(q′) [Lam73, V.3.15]. ♠

Let h_{K} be the standard 2-dimensional hyperbolic form h_{K}(x, y) = xy over an extension K of k discussed at the beginning of §13. For each ndimensional quadratic form q admitting a spin structure over K, denote ed_{n}(q) the essential dimension of the class of q in T_{n}(K).

Lemma 15.7. Let q be an dimensional quadratic form over K admitting a spin structure, and let s be a positive integer. Then we have

ed_{n+2s}(h_{K}^{⊕s} ⊕ q) ≥ ed_{n}(q) − \frac{s(s + 2n − 1)}{2}.

Proof. Set m =^{def} ed_{n+2s}(h_{K}^{⊕s} ⊕ q); let F be a field of definition of h_{K}^{⊕s} ⊕ q of transcendence degree m, and let q̃ be a quadratic form with a spin structure on F whose base change to K is isomorphic to h_{K}^{⊕s} ⊕ q. Let X be the Grassmannian of s-dimensional subspaces of F^{n+2s} which are totally isotropic with respect to q̃; the dimension of X is precisely s(s + 2n − 1)/2.

The variety X has a rational point over K; hence there exists an intermediate extension F ⊆ E ⊆ K such that tr deg_{F} E ≤ s(s + 2n − 1)/2, with the property that q̃_{E} has a totally isotropic subspace of dimension s. Then q̃_{E} splits as h_{E}^{s} ⊕ q′. By Witt’s cancellation Theorem, the base change of q′ to K is isomorphic to q; hence ed_{n}(q) ≤ m + s(s + 2n − 1)/2, as claimed. ♠

Proof of Theorem 15.4. If n ≤ 10 then the statement is vacuous, because then 2^{(n+4)/4} − n − 2 ≤ 0, so we assume that n ≥ 12. We may also assume without loss of generality that ζ_{4} ∈ k. In this case W(K) is a ℤ/2-vector space; it follows that the 3-Pfister number of a form q is the smallest r appearing in an expression

q = ∑_{i=1}^{r} ≪a_{i}, b_{i}, c_{i} ≫.

in W(K).

We will take an n-dimensional form q with a spin structure such that ed_{n}(q) = ed T_{n}. Suppose that q is equivalent in the Witt ring to a form of the type ∑_{1=1}^{r} ≪a_{i}, b_{i}, c_{i} ≫.

Let us write a Pfister form ≪a, b, c≫ as

≪a, b, c≫ = ⟨1⟩ ⊕ ≪a, b, c≫_{0},

where

≪a, b, c≫_{0} =^{def} ⟨a_{i}, b_{i}, c_{i}, a_{i}b_{i}, a_{i}c_{i}, b_{i}c_{i}, a_{i}b_{i}c_{i}⟩.

Set

ϕ =^{def} ∑_{1=1}^{r} ≪a_{i}, b_{i}, c_{i} ≫_{0}

if r is even, and

ϕ =^{def} ⟨1⟩ ⊕ ∑_{1=1}^{r} ≪a_{i}, b_{i}, c_{i} ≫_{0}

if r is odd. Then q is equivalent to ϕ in the Witt ring, and ϕ has a spin structure. The dimension of ϕ is 7r or 7r + 1, according to the parity of r.

We claim that n < 7r. If not, then the dimension of q is at most equal to the dimension of ϕ, so q is isomorphic to a form of type h_{K}^{s} ⊕ ϕ. By Lemma 15.6 and Theorem 1.13 we get the inequalities

\frac{3n}{7} ≥ 3r ≥ ed_{n}(q) = ed T_{n} ≥ ed Spin_{n} − 1 .

The resulting inequality fails for every even n ≥ 12 because, for such n, ed Spin_{n} ≥ n/2; see (1.14).

So we may assume that 7r ≥ n; then there is an isomorphism between the quadratic forms ϕ and a form of the type h_{K}^{⊕s} ⊕q. By comparing dimensions we get the equality 7r = n + 2s when r is even, and 7r + 1 = n + 2s when r is odd. The essential dimension of the form ϕ as an element of T_{7r}(K) or T_{7r+1}(K) is at most 3r, while by Lemma 15.7 we have that this essential dimension is at least ed_{n}(q) − s(s + 2n − 1)/2. From this, Lemma 15.5 and Theorem 1.13 we have the chain of inequalities

3r ≥ ed_{n}(q) − \frac{s(s + 2n − 1)}{2}

= ed T_{n} − \frac{s(s + 2n − 1)}{2}

≥ ed Spin_{n} − 1 − \frac{s(s + 2n − 1)}{2}

≥ 2^{(n−2)/2} − \frac{n(n − 1)}{2} − 1 − \frac{s(s + 2n − 1)}{2}.

Now assume that r is even. We substitute the value s = (7r − n)/2 in the resulting inequality, obtaining

\frac{49r^{2} + (14n + 10)r − 2^{(n+4)/2} − n^{2} + 2n − 8}{8} ≥ 0.

We interpret this as a quadratic inequality in r. The constant term of the polynomial is negative for all n ≥ 8; hence if r_{0} is the positive root, the equality is equivalent to r ≥ r_{0}. By the quadratic formula

r_{0} = \frac{\sqrt{49 · 2^{(n+4)/2} + 168n − 367} − (7n + 5)}{49}

≥ \frac{2^{(n+4)/4} − n − 2}{7} .

This completes the proof of Theorem 15.4 when r is even. The calculations when r is odd are analogous: using the substitution s = (7r + 1 − n)/2 we obtain the root

r_{0} = \frac{\sqrt{49 · 2^{(n+4)/2} + 168n − 199} − (7n + 12)}{49}

≥ \frac{2^{(n+4)/4} − n − 2}{7} .

♠

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(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