A GENERICITY THEOREM FOR ALGEBRAIC STACKS AND ESSENTIAL DIMENSION OF HYPERSURFACES

ZINOVY REICHSTEIN^{†} AND ANGELO VISTOLI^{‡}

Abstract. We compute the essential dimension of the functors Forms_{n,d} and Hypersurf _{n,d} of equivalence classes of homogeneous polynomials in n variables and hypersurfaces in ^{n−1} , respectively, over any base field k of characteristic 0. Here two polynomials (or hypersurfaces) over K are considered equivalent if they are related by a linear change of coordinates with coefficients in K. Our proof is based on a new Genericity Theorem for algebraic stacks, which is of independent interest. As another application of the Genericity Theorem, we prove a new result on the essential dimension of the stack of (not necessarily smooth) local complete intersection curves.

Contents

1. Introduction 1 2. Preliminaries 4 3. Amenable stacks and generic essential dimension 7 4. Gerbe-like stacks 9 5. The genericity theorem 10 6. Essential dimension of GL_{n}-quotients 16 7. Proof of Theorem 1.1 20 8. Small n and d 22 9. Essential dimension of singular curves 24 References 24

1. Introduction

Let k be a base field of characteristic 0, K/k be a field extension, and F(\underline{x}) be a homogeneous polynomial (which we call a form) of degree d in the n variables \underline{x} = (x_{1}, …, x_{n}), with coefficients in K. We say that F descends to an intermediate field k ⊂ K_{0} ⊂ K if there exists a linear change of coordinates g ∈ GL_{n}(K) such that every coefficient of F(g · \underline{x}) lies in K_{0}.

It is natural to look for a “smallest” subfield K_{0} to which a given form F(\underline{x}) descends. A minimal such field K_{0} with respect to inclusion may not exist, so we ask instead for the minimal transcendence degree trdeg_{k}K_{0}. This number, called

Date: March 19, 2011. 2000 Mathematics Subject Classification. Primary 14A20, 14J70. Key words and phrases. Essential dimension, hypersurface, genericity theorem, stack, gerbe. †Supported in part by NSERC Discovery and Accelerator Supplement grants. ‡Supported in part by the PRIN Project “Geometria sulle varietà algebriche”, financed by MIUR.

the essential dimension ed_{k}F of F, may be thought of as measuring the “complexity” of F. A major goal of this paper is to compute the maximum of ed_{k}F, taken over all fields K/k and all forms F(x_{1}, …, x_{n}) of degree d. This integer, usually called the essential dimension ed_{k}Forms_{n,d} of the functor of forms Forms_{n,d} depends only on n and d; it may be viewed as a measure of complexity of all forms of degree d in n variables.

We will also be interested in a variant of this problem, where the form F(\underline{x}) ∈ K[x_{1}, …, x_{n}] of degree d is replaced by the hypersurface

(1.1)  H =^{def} {(a_{1} : ⋯: a_{n}) | F(a_{1}, …, a_{n}) = 0}

in ^{n−1} . Here we say that H descends to K_{0} if there exists a linear change of coordinates g ∈ GL_{n}(K) and a scalar c ∈ K^{∗} such that every coefficient of cF(g · \underline{x}) lies in K_{0}. Once again, the essential dimension ed_{k}(H) of H is defined as the minimal value of trdeg_{k}K_{0}, with the minimum taken over all fields K_{0}/k such that H descends to K_{0}. We will be interested in the essential dimension ed_{k}(Hypersurf _{n,d}), defined as the maximal value of ed_{k}(H), where the maximum is taken over all K/k and all forms F(\underline{x}) ∈ K[x_{1}, …, x_{n}] of degree d. Here H is the zero locus of F, as in (1.1).

The study of forms played a central role in 19th century algebra. The problems of computing ed_{k}Forms_{n,d} and ed_{k}Hypersurf _{n,d} are quite natural in this context. However, to the best of our knowledge, these questions did not appear in the literature prior to the (relatively recent) work of G. Berhuy and G. Favi, who showed that ed_{k}Hypersurf _{3,3} = 3; see [BF04].

In this paper we compute ed_{k}Forms_{n,d} and ed_{k}Hypersurf _{n,d} for all n, d ≥ 1. Our main result is as follows.

Theorem 1.1. Assume that n ≥ 2 and d ≥ 3 are integers and (n, d) ≠ (2, 3), (2, 4) or (3, 3). Then

(a) ed_{k}Forms_{n,d} = \binom{n+d−1}{d} − n^{2} + cd(GL_{n}/μ_{d}) + 1 .

(b) ed_{k}Hypersurf _{n,d} = \binom{n+d−1}{d} − n^{2} + cd(GL_{n}/μ_{d}) .

The values of ed_{k}Forms_{n,d} and ed_{k}Hypersurf _{n,d} for n, d ≥ 1 not covered by Theorem 1.1 are computed in Section 8; the results are summarized in the following table.

n  d ed_{k}Forms_{n,d} ed_{k}Hypersurf _{n,d}
arbitrary 1  0  0
1  ≥ 2  1  0
arbitrary 2  n  n − 1
2  3  2  1
2  4  3  2
3  3  4  3

The quantity cd(GL_{n}/μ_{d}) which appears in the statement of Theorem 1.1 is the canonical dimension of the algebraic group GL_{n}/μ_{d}. For the definition and basic properties of canonical dimension we refer the reader to Section 2.2; see also [BR05, KM06] for a more extensive treatment of this notion. The exact value of cd(GL_{n}/μ_{d}) is known in the case where e =^{def} gcd(n, d) is a prime power p^{j}. In this case

cd(GL_{n}/μ_{d}) = \{ \begin{matrix}p^{i} − 1, if j > 0 , \\ 0, otherwise, \end{matrix}

where p^{i} is the highest power of p dividing n; see [BR05, Section 11]. More generally, suppose e = p_{1}^{j_{1}} …p_{r}^{j_{r}} is the prime decomposition of e (with j_{1}, …, j_{r} ≥ 1) and p_{s}^{i_{s}} is the highest power of p_{s} dividing n. A conjecture of J.-L. Colliot-Thélène, N. A. Karpenko, and A. S. Merkurjev [CTKM07, (2)] implies that

(1.2)  cd(GL_{n}/μ_{d}) = ∑_{s=1}^{r} (p_{s}^{i_{s}} − 1).

This has only been proved if e is a prime power (as above) or n = 6 [CTKM07, Theorem 1.3]. In these two cases Theorem 1.1 gives the exact value of ed_{k}Forms_{n,d} and ed_{k}Hypersurf _{n,d}. For other n and d Theorem 1.1 reduces the problems of computing ed_{k}Forms_{n,d} and ed_{k}Hypersurf _{n,d} to the problem of computing the canonical dimension cd(GL_{n}/μ_{d}). For partial results on the latter problem, see [BR05, Section 11].

The notions of essential dimension for forms and hypersurfaces are particular cases of Merkurjev’s general definition of essential dimension of a functor [BF03]. A special case of this, upon which our approach is based, is the essential dimension of an algebraic stack. For background material on this notion we refer the reader to [BRV09]. In particular, ed_{k}Forms_{n,d} = ed_{k}[A_{n,d}/GL_{n}] and ed_{k}Hypersurf _{n,d} = ed_{k}[ (A_{n,d})/GL_{n}], where A_{n,d} is the \binom{n+d−1}{d}-dimensional affine space of forms of degree d in n variables and (A_{n,d}) is the associated \binom{n+d−1}{d} − 1 dimensional projective space of degree d hypersurfaces in ^{n−1} . (Here, as in the rest of the paper, we will follow the classical convention of defining the projectivization (V) of a vector space V over k as the projective space of lines in V, that is, as ProjSym_{k}V^{∨}. In the present context, this seems more natural than Grothendieck’s convention of defining (V) as ProjSym_{k}V.) The group GL_{n} naturally acts on these spaces, and [A_{n,d}/GL_{n}] and [ (A_{n,d})/GL_{n}] denote the quotient stacks for these actions; see [BRV09, Example 2.6].

The essential dimension of the “generic hypersurface” of degree d is ^{n−1} , i.e., of the hypersurface H_{gen} cut out by the “generic form”

(1.3)  F_{gen}(x_{1}, …, x_{n}) = ∑_{i_{1}+··· +i_{n}=d} a_{i_{1},...,i_{n}}x_{1}^{i_{1}} …x_{n}^{i_{n}} = 0 ,

where a_{i_{1},...,i_{n}} are independent variables and K is the field generated by these variables over k, was computed in [BR05, Sections 14-15]. The question of computing the essential dimension of the generic form F_{gen} itself was left open in [BR05]. For n and d as in Theorem 1.1 we will show that ed_{k}F_{gen} = ed_{k}H_{gen} + 1; see Proposition 3.4.

The key new ingredient in the proof of Theorem 1.1 is the following “Genericity Theorem”. Let X be a connected algebraic stack with quasi-affine diagonal that is smooth of finite type over k, in which the automorphism groups are generically finite (for sake of brevity, we say that X is amenable). Then we can define the generic essential dimension of X, denoted by ged_{k}X, as the supremum of the essential dimensions of the dominant points Spec K → X. If X is Deligne–Mumford, that is, if all stabilizers are finite, then ed_{k}X = ged_{k}X; see [BRV09, Theorem 6.1].

This result, which we called the Genericity Theorem for Deligne–Mumford stacks in [BRV09], is not sufficient for the applications in the present paper. Here we prove the following stronger theorem conjectured in [BRV09, Question 6.6].

Theorem 1.2. Let X be an amenable stack over k. Let L be a field extension of k, and ξ be an object of X (Spec L), such that the automorphism group scheme \underline{Aut}_{L} ξ is reductive. Then

ed_{k}ξ ≤ ged_{k}X .

In particular, if the automorphism group of any object of X defined over a field is reductive, then ed_{k}X = ged_{k}X.

Note that Theorem 1.2 fails if the stabilizers are not required to be reductive (see [BRV09, Example 6.5(b)]), even though a weaker statement may be true in this setting (see Conjecture 5.3). We also remark that the locus of points with reductive stabilizer is constructible but not necessarily open in X. Thus for the purpose of proving Theorem 1.2 it does not suffice to consider the case where all stabilizers are reduced.

Theorem 1.2 implies, in particular, that if the automorphism group of a form f(x_{1}, …, x_{n}) is reductive then ed_{k}f ≤ ed_{k}F_{gen}. To complete the proof of Theorem 1.1(a) we supplement this inequality with additional computations, carried on in Section 6, which show that forms f(x_{1}, …, x_{n}) whose automorphism group is not reductive have low essential dimension; for a precise statement, see Theorem 6.3. The proof of Theorem 1.1(b) is more delicate because the quotient stack [ (A_{n,d})/GL_{n}] is not amenable, so the Genericity Theorem cannot be applied to it directly. We get around this difficulty in Section 7 by relating ed_{k}[ (A_{n,d})/GL_{n}] to the essential dimension of the amenable stack [ (A_{n,d})/PGL_{n}].

In the last section we use our Genericity Theorem 1.2 to prove a new result on the essential dimension of the stack of (not necessarily smooth) local complete intersection curves, strengthening [BRV09, Theorem 7.3].

Acknowledgments. We are grateful to J. Alper and P. Brosnan for helpful discussions.

2. Preliminaries

2.1. Special groups. A linear algebraic group scheme G over k is said to be special if for every extension K/k we have H^{1}(K, G) = {1}. Special groups were studied by Serre [CAS58, Exposé 1] and classified by Grothendieck [CAS58, Exposé 5] (over an algebraically closed field of characteristic 0). Note that G is special if and only if ed_{k}G = 0; see [TV10, Proposition 4.3].

The group GL_{n} is special by Hilbert’s Theorem 90, and so is the special linear group SL_{n}. Direct products of special groups are easily seen to be special. Moreover, in characteristic 0 the group G is special if and only if the Levi subgroup of G (which is isomorphic to G/ R_{u}G) is special; see [San81, Theorem 1.13]. Here R_{u}G denotes the unipotent radical of G. We record the following fact for future reference.

Let A be a non-zero nilpotent n × n-matrix with entries in k and G_{A} be the image of the map _{a} → GL_{n} given by t → exp(tA). Note that this map is algebraic, since only finitely many terms in the power series expansion of exp(tA) are non-zero.

Lemma 2.1.

(a) The centralizer C of A (or equivalently, of G_{A}) in GL_{n} is special.

(b) The normalizer N of G_{A} in GL_{n} is special.

Proof. (a) By [Jan04, Propositions 3.10 and 3.8.1] C is a semidirect product U ⋊ H, where U ◁ C is unipotent and H is the direct product of general linear groups GL_{r} for various r ≥ 0; cf., also [McN06, Section 2]. Thus H = Levi subgroup of C is special, and part (a) follows.

(b) The normalizer N acts on G_{A} ≃ _{a} by conjugation. This gives rise to a homomorphism π: N → _{m} = \underline{Aut}_{k} _{a} whose kernel is the centralizer C = C_{GL_{n}}(A). If π is trivial then N = C is special by part (a). If π is non-trivial then it is surjective, and we have an exact sequence

1 −→ C −→ N −→ _{m} −→ 1 .

The long non-abelian cohomology sequence for H^{0} and H^{1} associated with this short exact sequence shows that H^{1}(K, N) = {1} for every field K/k, as desired.

♠

2.2. Canonical dimension. Let K be a field and X be either a geometrically integral smooth complete K-scheme of finite type or a G-torsor for some connected linear algebraic K-group G. The canonical dimension cd X of X is the minimal value of dim Y, where Y ranges over all integral closed K-subschemes of X admitting a rational map X −−→ Y defined over K. Equivalent definitions via generic splitting fields and determination functions can be found in [BR05, KM06].

If we fix a base field k and an algebraic k-group G, the maximal value of cd X as K ranges over all field extensions K/k and X → Spec K ranges over all G_{K}- torsors, is denoted by cd G. Moreover, cd G = cd X_{ver}, where X_{ver} → Spec K_{ver} is a versal G-torsor. In particular, we can construct a versal G-torsor by starting with a generically free linear representation V of G defined over k and setting K_{ver} =^{def} k(V)^{G}. Then V has a G-invariant open subset U which is the total space of a G-torsor U → B, where k(B) = k(V)^{G}. Restricting to the generic point η of B, we obtain a versal torsor X_{ver} =^{def} U_{η} → Spec K_{ver}. For details of this construction we refer the reader to [GMS03, I.5].

Lemma 2.2.

(a) Let X_{1} and X_{2} be Brauer-equivalent Brauer–Severi varieties over a field K/k. Then

cd X_{1} = cd X_{2} .

In other words, the canonical dimension cd α of a Brauer class α ∈ H^{2}(K, _{m}) is well defined.

(b) Let G = GL_{n} or SL_{n} and let C be a central subgroup scheme of G. Then for any field K/k and any (G/C)-torsor X → Spec K we have cd X = cd α, where α is the image of the class of X under the coboundary map ∂_{K}: H^{1}(K, G/C) → H^{2}(K, C) ⊆ H^{2}(K, _{m}) induced by the exact sequence 1 → C → G → G/C → 1.

(c) Let K/k be a field extension and α ∈ H^{2}(K, _{m}) be a Brauer class of index dividing n and exponent dividing d. Then cd α ≤ cd(GL_{n}/μ_{d}).

Proof. (a) follows from the fact that X_{1} and X_{2} have the same splitting fields L/K; see [BR05, Section 10] or [KM06, Section 2].

(b) By Hilbert’s Theorem 90, G is special, i.e., H^{1}(L, G) = {1} for any field L. Hence, the coboundary map ∂_{L}: H^{1}(L, G/C) → H^{2}(K, C) has trivial kernel for any L, and the desired conclusion follows from [BR05, Lemma 10.2].

(c) By our assumption, α lies in the image of the coboundary map

∂_{K}: H^{1}(K, GL_{n}/μ_{d}) −→ H^{2}(K, C) ;

cf., e.g., [BR05, Lemma 2.6]. Part (c) now follows from part (b). ♠

The following result will be used repeatedly in the sequel.

Proposition 2.3.

(a) Let X → Spec K be a _{m}-gerbe over a field K. Denote the class of this gerbe in H^{2}(K, _{m}) by α. Then ed_{K}X = cd α.

(b) Let e ≥ 1 be an integer and X → Spec K be a μ_{e}-gerbe over a field K. Denote the class of this gerbe in H^{2}(K, μ_{e}) by β. Then ed_{K}X = cd β + 1.

Proof. See [BRV09, Theorem 4.1]. ♠

2.3. Gerbes and Brauer classes. Let ϕ: X → \overline{X} be a _{m}-gerbe over a stack \overline{X}. If L is a field and ξ ∈ \overline{X}(L) then, pulling back X to Spec L we obtain a _{m}-gerbe X_{ξ} over L. We will denote by ind(X_{ξ}) and exp(X_{ξ}) the index and exponent of the Brauer class of X_{ξ}. The following lower semi-continuity properties of ind and exp (as functions of ξ) will be used in the proof of Theorem 1.1(b).

Lemma 2.4. Let ϕ: X → \overline{X} be a _{m}-gerbe over an integral regular algebraic stack \overline{X}, as above. Assume further that \overline{X} is generically a scheme, with generic point η: Spec K → \overline{X}. Then for any field L/k and any ξ ∈ \overline{X}(L),

(a) ind(X_{ξ}) divides ind(X_{η}), and

(b) exp(X_{ξ}) divides exp(X_{η}).

Proof. (a) The key fact we will use is that if B is a Brauer–Severi variety over a field L then ind(B) divides d if and only if B has a linear subspace of dimension d − 1 defined over L; see [Art82, Proposition 3.4].

By [LMB00a, Theorem 6.3] there exists a smooth map T → X such that ξ lifts to a point Spec L → T; we may assume that T is affine and integral. The index of the pullback of X_{η} to the function field k(T) divides ind(X_{η}); hence we can substitute \overline{X} with T, and assume that \overline{X} = T is an affine regular integral variety. The étale cohomology group H^{2}(T, _{m}) is torsion, because T is regular; hence, by a well known result of O. Gabber [Gab81] the class of X is represented by a Brauer–Severi scheme P → T.

Let d be the index ind(\overline{X}_{η}) =^{def} ind(P_{η}) and Gr(P, d − 1) → T be the Grassmannian bundle of linear subspaces of dimension d − 1 in P. The generic fiber Gr(P, n − 1)_{η} has a K-rational point; this gives rise to a section U → Gr(P, n − 1) over some open substack U of T. Let Y be the complement of U in T. If our point ξ: Spec L → \overline{X} lands in U, then the pullback P_{ξ} has a linear subspace of dimension d − 1 defined over L, and we are done. Thus we may assume that ξ ∈ Y(L). The morphism ξ: Spec L → T extends to a morphism Spec R → T, where R is a DVR with residue field L, such that the generic point of Spec R lands in the complement of Y in T. The pullback Gr(P, d − 1)_{R} of Gr(P, d − 1) to Spec R then has a section over the generic point. By the valuative criterion of properness this section extends to a section Spec R → Gr(P, d − 1). Specializing to the closed point of Spec R, we obtain a desired section Spec L → Gr(P, d − 1). This shows that P_{ξ} has degree dividing d, as claimed.

(b) Set e =^{def} exp(X_{ξ}) and apply part (a) to the e^{th} power Y of the gerbe X. Since Y_{η} is trivial (i.e., has index 1), so is Y_{ξ}. But Y_{ξ} is the e^{th} power of the class of X_{ξ}, and we are done.

An alternative proof of part (b) is based on the fact that a Brauer-Severi variety B → Spec L over a field L has index dividing e if and only if P contains a hypersurface of degree e defined over L; see [Art82, (5.2)]. We may thus proceed exactly as in the proof of part (a), with the same T and P → T, but using the Hilbert scheme H(P, e) → T of hypersurfaces of degree e in P instead of the Grassmannian. ♠

3. Amenable stacks and generic essential dimension

Definition 3.1. Let X be an algebraic stack over k. We say that X is amenable if the following conditions hold.

(a) X is integral with quasi-affine diagonal.

(b) X is locally of finite type and smooth over k.

(c) There exists a non-empty open substack of X that is a Deligne–Mumford stack.

Any irreducible algebraic stack has a generic gerbe, the residual gerbe at any dominant point Spec K → X [LMB00a, § 11]. For amenable stacks, there is an alternate description. Let X be an amenable stack over k, and U a non-empty open substack which is Deligne–Mumford. After shrinking U, we may assume that the inertia stack I_{U} is finite over U. Let U be the moduli space of U, whose existence is proved in [KM97], and let k(X) be its residue field. The generic gerbe X_{k(X)} → Spec k(X) is then the fiber product Spec k(X) ×_{U} U. The dimension dim X is the dimension of U, or, equivalently, the dimension of U.

Example 3.2. Consider the action of a linear algebraic group defined over k on a smooth integral k-scheme X, locally of finite type. Then the quotient stack [X/G] is amenable if and only if the stabilizer Stab_{G}(x) of a general point x ∈ X is finite.

Of particular interest to us will be the GL_{n}-actions on A_{n,d}, the \binom{n+d−1}{d}-dimen-sional affine space of forms of degree d in n variables, and (A_{n,d}) = \binom{n+d−1}{d} − 1 dimensional projective space of degree d hypersurfaces in ^{n−1} , as well as the PGL_{n}-action on (A_{n,d}).

Since the center of GL_{n} acts trivially on (A_{n,d}), the stack [ (A_{n,d})/GL_{n}] is not amenable. On the other hand, it is classically known that the stabilizer of any smooth hypersurface in ^{n−1} of degree d ≥ 3 is finite; see, e.g., [OS78, Theorem 2.1] or [MM64]. From this we deduce that the stacks [ (A_{n,d})/PGL_{n}] and [A_{n,d}/GL_{n}] are both amenable for any n ≥ 2 and d ≥ 3.

Moreover, if n ≥ 2, d ≥ 3 and (n, d) ≠ (2, 3), (2, 4) or (3, 3) then the stabilizer of a general hypersurface in ^{n−1} of degree d is trivial; see [MM64]. For these values of n and d the quotient stack [ (A_{n,d})/PGL_{n}] is generically a scheme of dimension

dim (A_{n,d}) − dim PGL_{n} = \binom{n + d − 1}{d} − n^{2} .

Definition 3.3. The generic essential dimension of an amenable stack X is

ged_{k}X =^{def} ed_{k(X)}X_{k(X)} + dim X .

Alternatively, ged_{k}X is the supremum of the essential dimension of ζ ∈ X (K), taken over all field extensions K/k and all dominant ζ: Spec K → X. By the Genericity Theorem for Deligne–Mumford stacks [BRV09, Theorem 6.1], we see that ged_{k}X is the essential dimension of any open substack of X that is a Deligne– Mumford stack.

We will now compute the generic essential dimension of the quotient stacks [A_{n,d}/GL_{n}] and [ (A_{n,d})/GL_{n}] for n and d as in the statement of Theorem 1.1. Recall that ged_{k}[A_{n,d}/GL_{n}] = ed_{k}F_{gen} and ged_{k}[ (A_{n,d})/GL_{n}] = ed_{k}H_{gen}, where F_{gen} is the generic forms of degree d in n variables and H_{gen} is the generic hypersurfaces, as in (1.3).

Proposition 3.4. Let n ≥ 2 and d ≥ 3 be integers. Assume further that (n, d) ≠ (2, 3), (2, 4) or (3, 3). Then

(a) ged_{k}[ (A_{n,d})/GL_{n}] = \binom{n+d−1}{d} − n^{2} + cd(GL_{n}/μ_{d}).

(b) ged_{k}[A_{n,d}/GL_{n}] = \binom{n+d−1}{d} − n^{2} + cd(GL_{n}/μ_{d}) + 1.

Part (a) was previously known; see [BR05, Theorem 15.1]. Part (b) answers an open question from [BR05, Remark 14.8].

Proof. Let X = [ (A_{n,d})/GL_{n}], Y = [A_{n,d}/GL_{n}], and \overline{X} = [ (A_{n,d})/PGL_{n}]. Consider the diagram

\xymatrix{ & Y_{η} \ar[dl] \ar[ddl] & Y \ar[dl] \ar[ddl]^{μ_{d}-gerbe} \\ X_{η} \ar[d] & X \ar[d]_{_{m}-gerbe} \\ η \ar[r] & \overline{X}}

For n and d as in the statement of the proposition, [ (A_{n,d})/PGL_{n}] is generically a scheme (see Example 3.2). Denote the generic point of this scheme by η and its function field by k(η). The pull-backs Y_{k(η)} and X_{k(η)} are, respectively, a μ_{d}- gerbe and a _{m}-gerbe over k(η); these two gerbes give rise to the same class α ∈ H^{2}(k(η), μ_{d}) ⊂ H^{2}(k(η), _{m}). By Proposition 2.3

ed_{k(η)} Y_{k(η)} = cd α and ed_{k(η)} X_{k(η)} = cd α + 1.

Since

trdeg_{k}k(η) = \binom{n + d − 1}{d} − n^{2} ,

it remains to show that

(3.1)  cd α = cd(GL_{n}/μ_{d}) .

The action of G = GL_{n}/μ_{d} on A_{n,d} is linear and generically free. Thus it gives rise to a versal G-torsor t ∈ H^{1}(k(η), G), and α is the image of t under the natural coboundary map H^{1}(k(η), G) → H^{2}(k(η), μ_{d}) associated with the exact sequence 1 → μ_{d} → GL_{n} → G → 1. As we explained in Section 2.2, cd t = cd(GL_{n}/μ_{d}). On the other hand, by Lemma 2.2(b), cd α = cd t, and (3.1)follows. ♠

4. Gerbe-like stacks

The purpose of the next two sections is to prove the Genericity Theorem 1.2. The proof of the genericity theorem for Deligne–Mumford stacks in [BRV09] relied on a stronger form of genericity for gerbes; see [BRV09, Theorem 5.13]. Our proof of Theorem 1.2 will follow a similar pattern, except that instead of working with gerbes we will need to work in the more general setting of gerbe-like stacks, defined below. The main result of this section, Theorem 4.6, is a strong form of genericity for gerbe-like stacks.

Definition 4.1. A Deligne–Mumford stack X is gerbe-like if its inertia stack I_{X} is étale over X.

If X is an algebraic stack, the gerbe-like part X^{0} of X is the largest open substack of X that is Deligne–Mumford and gerbe-like.

Remark 4.2. If an algebraic stack X is Deligne–Mumford, then the inertia stack I_{X} → X is unramified. Hence, if X is also reduced then by generic flatness the gerbe-like part X^{0} of X is dense in X.

Lemma 4.3. Let X be a reduced Deligne–Mumford stack. Suppose that the inertia stack I_{X} is finite and étale over X. Then X is a proper étale gerbe over an algebraic space.

Remark 4.4. The condition that X be reduced can be eliminated. However, it makes the proof marginally simpler, and will be satisfied in all cases of interest to us in this paper.

Proof. Let X be the moduli space of X; we claim that X is a proper étale gerbe over X. This is a local problem in the étale topology of X. Hence, after passing to an étale covering of X, we may assume that X is a connected scheme, and there exists a finite reduced connected scheme U, with a finite group G acting on U, such that X = [U/G]. The pullback of I_{X} to U is the closed subscheme of G × U defined as representing the functor of pairs (g, u) with gu = u. The fact that this pullback is étale over U translates into the condition that the order of the stabilizer of a geometric point is locally constant on U. Since U is connected, this means that there exists a subgroup H of G that is the stabilizer of all the geometric points of G; this subgroup is necessarily normal. The induced action of (G/H) is free, and U/(G/H) = X; hence U is étale over X, and X = [U/G] is a gerbe banded by H over X. ♠

Lemma 4.5. Suppose that X is a gerbe-like Deligne–Mumford stack, Y → X a representable unramified morphism. Then Y is also gerbe-like.

Proof. The inertia stack I_{X} of a stack X is the fiber product X ×_{X ×X} X. We have a diagram

[figure: Y ×_{X} Y ×_{X} I_{X}; Y ×_{X} Y; I_{X}; X; X; X × X; Y ×_{X} Y; Y × Y]

in which all the squares are cartesian. This implies the equality

Y ×_{X} Y ×_{X} I_{X} = (Y ×_{X} Y) ×_{X} (Y ×_{X} Y) ,

which in turn tells us that (Y ×_{X} Y) ×_{X} (Y ×_{X} Y) is étale over Y ×_{X} Y. The hypotheses on Y → X imply that the diagonal Y → Y ×_{X} Y is an open embedding. Thus I_{Y} = Y_{Y×Y} is an open substack of (Y ×_{X} Y) ×_{X} (Y ×_{X} Y), so it is étale over Y ×_{X} Y hence it is étale over Y, as claimed. ♠

From this and the results in [BRV09], it is easy to deduce the following. Given a field L/k and ξ ∈ X (Spec L), we denote by codim_{X} ξ the codimension of the closure of the image of the corresponding morphism Spec L → X.

Theorem 4.6. Let X be an integral gerbe-like Deligne–Mumford stack which is smooth of finite type over a field k. Let L be an extension of k and ξ ∈ X (Spec L). Then

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

Proof. If the inertia stack I_{X} is finite over X, then, by Lemma 4.3, X is an étale proper gerbe over a smooth k-scheme, and the statement reduces to [BRV09, Theorem 5.13]. In the general case, from [BRV09, Lemma 6.4] we deduce the existence of an étale representable morphism Y → X, such that Y is an integral Deligne– Mumford stack with finite inertia, and the morphism Spec L → Y factors through Y. By Lemmas 4.3 and 4.5, the stack Y is a proper étale gerbe over a smooth algebraic space, hence [BRV09, Theorem 5.13] can be applied to it. Let η ∈ Spec Y be a point in Y(L) mapping to ξ. Then we have the relations

ed_{k}ξ ≤ ed_{k}η,
ed_{k(X)} X_{k(X)} ≥ ed_{k(Y)} and
codim_{X} ξ = codim_{Y} η;

hence the general case of the Theorem follows from [BRV09, Theorem 5.13]. ♠

5. The genericity theorem

We now proceed with the proof of Theorem 1.2. As before, let Y be the closure of the image of ξ: Spec L → X. The stack Y is integral, and since chark = 0, Y is generically smooth. Let π: M → _{k}^{1} the deformation to the normal bundle of Y inside X; then π^{−1} ( _{k}^{1}∖ {0}) = X ×_{Spec k}( _{k}^{1}∖ {0}), while π^{−1} (0) is isomorphic to the normal bundle N of Y in X.

Lemma 5.1. M^{0} ∩ N ≠ ∅.

Theorem 1.2 follows from Lemma 5.1 and Theorem 4.6 by the same argument as in [BRV09, Theorem 6.1]. This argument is quite short, and we reproduce it here for sake of completeness.

Let L be an extension of k and let ξ be an object of X (L). Call Y the closure of the image of the morphism ξ: Spec L → X, with its reduced stack structure. Set N^{0} =^{def} M^{0}∩ N. Then the fiber product Spec L ×_{Y} N is a vector bundle over Spec L, and Spec L ×_{X} M^{0} is a non-empty open subscheme. Hence ξ: Spec L → Y can be lifted to N^{0}; this gives an object η of N^{0}(Spec L) mapping to ξ in Y. Clearly the essential dimension of ξ as an object of X is the same as its essential dimension as an object of Y, and ed_{k}ξ ≤ ed_{k}η. Let us apply Theorem 4.6 to the gerbe M^{0}. The function field of the moduli space M of M is k(X)(t), and its generic gerbe is X_{k(X)(t)}; by [BRV09, Proposition 2.8], we have ed_{k(X)(t)} X_{k(X)(t)} ≤ ed_{k(X)}X_{k(X)}. The composite Spec L → N^{0} ⊆ M^{0} has codimension at least 1, hence we obtain

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

This concludes the proof.

Proof of Lemma 5.1.If X is a finite dimensional representation of a group scheme G over a field k, we will identify X with the affine space Spec(Sym_{k}^{•} X^{∨}).

Let us suppose that X is a finite dimensional representation of a linearly reductive algebraic group G and Y ⊆ X is a subrepresentation. Since G is reductive, we have a G-equivariant splitting X ≃ Y ⊕ Y′. Set X =^{def} [X/G] and Y =^{def} [Y/G]. Assume that the generic stabilizer of the action of G on X is finite. Then X is amenable, and Y ⊆ X is a closed integral substack.

It is easy to see that the deformation to the normal bundle M of Y in X is Gequivariantly isomorphic to X ×_{k} _{k}^{1} (where the group G acts trivially on _{k}^{1}); the projection

Y × Y′×_{k} _{k}^{1} = X ×_{k} _{k}^{1} ≃ M −→ X ×_{k} _{k}^{1}

is given by the formula (y, y′, t) ↦ (ty, y′, t). The deformation to the normal bundle M of Y is [M/G] = [X/G] ×_{k} _{k}^{1}; hence M^{0} = X^{0} ×_{k} _{k}^{1}, and it is obvious that M^{0} ∩ N ≠ ∅.

The proof in the general case will be reduced to this by a formal slice argument.

We may base-change to the algebraic closure of k; so we may assume that k is algebraically closed. By deleting the singular locus of Y, we may assume that Y is smooth; by further restricting, we may assume that the inertia stack I_{Y} is flat over Y, and that all geometric fibers are reductive, and have the same numbers of connected components.

Let y_{0} → Spec k → Y be a general closed point. The residual gerbe G_{y_{0}} → Spec k [LMB00a, § 11] admits a section Spec k → G_{y_{0}}, since k is algebraically closed; hence,

if G =^{def} \underline{Aut}_{k}y_{0} it the automorphism group scheme of y_{0}, we have G_{y_{0}} ≃ B_{k}G. The embedding of stacks B_{k}G → Y is of finite type; hence it is easy to see that it is a locally closed embedding, from Zariski’s main Theorem for stacks [LMB00a, Théorème 16.5].

Let U → X be a smooth morphism, where U is a scheme, together with a lifting u_{0}: Spec k → U of y_{0}. Call X_{n} the n^{th} infinitesimal neighborhood of B_{k}G inside X: in other words, if U is an open substack of X containing B_{k}G as a closed substack, and we denote by I the sheaf of ideals of B_{k}G inside U, then X_{n} is the closed substack of X defined by the sheaf of ideals I^{n+1} . In particular, X_{0} = B_{k}G.

Lemma 5.2. There exists a finite dimensional representation X of G with finite generic stabilizers and a trivial subrepresentation Y ⊆ X, with the following property. If we denote by X_{n} and Y_{n} the n^{th} infinitesimal neighborhoods of the origin, there is a sequence of isomorphisms X_{n} ≃ [X_{n}/G], compatible with the embeddings X_{n} ⊆ X_{n+1} and [X_{n}/G] ⊆ [X_{n+1} /G], that induce isomorphisms of Y_{n} with [Y_{n}/G].

Furthermore, denote by X̂ the spectrum of the completion of the local ring of X at the origin. Then there exists a smooth morphism U → X with a closed point u_{0} ∈ U mapping to y_{0} in Y, and an isomorphism of X̂ with the spectrum Û of the completion of the local ring of U at u_{0}, such that (a) the sequence of composites X_{n} → [X_{n}/G] ≃ X_{n} ⊆ X is obtained by restriction from

the composite morphism X̂ ≃ Û → U → X, and

(b) the inverse image of Y in X̂ corresponds to the inverse image of Y in Û.

Proof. The tautological G-torsor P_{0} =^{def} Spec k → B_{k}G extends to a G-torsor P_{n} → X_{n}, in such a way that the restriction of P_{n+1} to X_{n} ⊆ X_{n+1} is isomorphic to P_{n}, by [Alp09, Propositions 4.1 and 4.2]. Each of the stacks P_{n} is in fact a scheme, because its reduced substack is; in fact, P_{n} must be the spectrum of a local artinian k-algebra R_{n}. Clearly, X_{n} = [P_{n}/G].

If we denote by V the maximal ideal of R_{1}, then R_{1} = k ⊕ V; the action of G on R_{1} induces a linear action of G on V. The space V is isomorphic to I/I^{2}, which is a coherent sheaf on B_{k}G, i.e., a representation of G. In turn, I/I^{2} is the cotangent space of deformations of Spec k → X, that is, the dual to the space of isomorphism classes of liftings Spec k[ϵ] → X of Spec k → X (here k[ϵ] denotes, as usual, the ring of dual numbers k[x]/(x^{2})).

The homomorphism R_{n+1} → R_{n} induced by the embedding P_{n} ≃ P_{n+1} |_{X_{n}} ⊆ P_{n+1} is surjective; its kernel is the ideal I^{n}R_{n+1} . Denote by R the projective limit \underleftarrow{lim}_{n} R_{n}. (Notice that, while G acts, by definition, on each of R_{n}, this does not, unfortunately, induced an action of G on R, as an algebraic group, unless G is finite; it if did, this would make the proof conceptually much simpler.) If x_{1}, …, x_{n} is a set of elements of R that project to a basis for V in R_{1}, the ring R is a quotient of the power series ring k[[x_{1}, …, x_{n}]] by an ideal J contained in 𝔪_{R}^{2}. We claim that J = 0, i.e., R is a power series ring. For this, it is enough to check that R is formally smooth over k, or, in other words, that if A is a local artinian k-algebra with reside field k and B is a quotient of A, any homomorphism of k-algebras R → B lifts to a homomorphism R → A. Take n ≫ 0; then R → B factors through R_{n}. Consider the composite

Spec B −→ Spec R_{n} −→ X_{n} ⊆ X ;

since X is smooth, deformations are unobstructed, i.e., this morphism extends to Spec A → X. If n ≫ 0, this factors as Spec A → X_{n} ⊆ X; and since Spec R_{n} is smooth over X_{n}, as it is a G-torsor, the section Spec B → Spec R_{n} of Spec B → X_{n} lifts to a section Spec A → R_{n}, giving the desired extension R → R_{n} → A.

Suppose that U → X a smooth morphism, where U is a scheme, with a lifting u_{0}: Spec k → U of y_{0}. Let us assume that U is minimal at u_{0}, or, in other words, that the tangent space of U at u_{0} maps isomorphically onto the deformation space of X at y_{0}. Since U is smooth over X, the morphisms Spec R_{n} → X lift to a compatible system of morphism Spec R_{n} → U, sending Spec k into u_{0}; these yield a morphism Spec R → O_{U,u_{0}}, inducing a homomorphism of k-algebras Ô_{U,u_{0}} → R. This is a homomorphism of power series algebras over k which gives an isomorphism of tangent spaces; hence it is an isomorphism. This shows that the morphisms Spec R_{n} → X yield a flat morphism Spec R → X.

Call d the codimension of Y in X at the point y_{0}; after a base change in R = k[[x_{1}, …, x_{n}]], we may assume that the inverse image of the ideal of Y in X is (x_{1}, …, x_{d}). Call X the scheme corresponding to the dual of the vector space V = ⟨x_{1}, …, x_{n}⟩, that is, X =^{def} Spec Sym_{k}^{•} V; call Y the linear subscheme defined by the ideal (x_{1}, …, x_{d}). Then X̂ = Spec R; the representation X has all the required properties, except that we have not yet proved that the action of G on X has finite generic stabilizers, and the representation Y is trivial.

To do this, let us call I the pullback of the inertia stack of [X/G] to X; in other words, I is the subscheme of G ×_{Spec k}X defined by the equation gx = x. We need to show that I is generically finite over X. Since I is a group scheme over X, it has equidimensional fibers, hence it is enough that there is an étale neighborhood I′→ I of the pair (1, u_{0}) in I which is generically finite over X. The inverse image of X_{n} in U is the n^{th} infinitesimal neighborhood U_{n} of u_{0} in U. Denote by J the pullback of the inertia stack of X to U; we have isomorphisms X_{n} ≃ U_{n}, and compatible isomorphisms of the pullbacks of I and J to X_{n} and U_{n} respectively. These induce an isomorphism of the completions of I and J at (1, x_{0}) and (1, u_{0}) respectively; by Artin approximation, the morphisms I → X and J → U are étalelocally equivalent at (1, x_{0}) and (1, u_{0}). Since J is generically finite over U it follows that the action of G on X has generically finite stabilizers. Also, this implies that the stabilizer of a general closed point of Y is isomorphic to the isomorphism group scheme of a general point of Spec k → Y, hence it has the same dimension and the same number of connected components of G; hence it equals G, and so the action of G on Y is trivial, as claimed. ♠

Consider the complement F of (M ∖ N)^{0} in M ∖ N. Since N is a divisor on M, the closure of F in M will not contain it; hence, if v is a general k-rational point of N, this has neighborhood in which the only point of the inertia stack I_{M} where this can fail to be étale are over N; so it is enough to show that I_{M} is étale over M at a general point of N. For this it suffices to show that the locus of point of the inverse image I_{N} of N in I_{M} at which I_{M} is étale over M surjects onto Y.

Denote by N the normal bundle of Y in X, and by M the deformation to the normal bundle. If n_{0} is a general closed point of N, then the pullback I_{M} of the inertia stack of [M/G] to M is étale at n_{0}. Notice that I_{M} is étale at a general closed point of the fiber of N over any y_{0} ∈ Y(k), since the action of G on Y is trivial, so translation by any closed point of Y is G-equivariant. Denote by M_{n} the inverse image of X_{n} ×_{Spec k} _{k}^{1} in M, and by M_{n} the inverse image of X_{n} ×_{Spec k} _{k}^{1} in M.

Claim. There is sequence of isomorphisms M_{n} ≃ [M_{n}/G] compatible with the isomorphisms X_{n} ≃ [X_{n}/G], and with the identity on _{k}^{1}.

Let us assume the claim. We know that I_{[M/G]} is étale over M at a general k-rational point of I_{[M/G]} lying over the image of the origin in [N/G] ⊆ [M/G]; let v ∈ I_{M} (k) be a general k-rational point lying over N, and let us show that I_{M} → M is étale at v. For this we use the infinitesimal criterion for étaleness. Let A be a finite k-algebra with residue field equal to k, let I be a proper ideal in A, and consider a commutative diagram

\xymatrix{Spec(A/I) \ar[r] \ar[d] & I_{M} \ar[d] \\ Spec A \ar[r] \ar@{-->}[ur] & M}

in which composite Spec k ⊆ Spec A → I_{M} is isomorphic to v; we need to show that we can fill in the dashed arrow in a unique way. For n ≫ 0, the morphism Spec A → M factor through M_{n}. Since I_{M_{n}} = M_{n} ×_{M} I_{M} , the square above factors through a square

\xymatrix{Spec(A/I) \ar[r] \ar[d] & I_{M_{n}} \ar[d] \\ Spec A \ar[r] \ar@{-->}[ur] & M_{n}}

in which again we have to show the existence and uniqueness of the lifting. However, the isomorphism M_{n} ≃ [M_{n}/G] induces an isomorphism of the morphism I_{M_{n}} → M_{n} with I_{[M_{n}/G]} → [M_{n}/G]; we know that I_{[M_{n}/G]} = [M_{n}/G] ×_{[M/G]} I_{[M/G]} is étale at the point corresponding to v. Hence the lifting exists and is unique.

Now let us prove the claim. Set

R =^{def} U ×_{X} U,
R_{n} =^{def} U_{n} ×_{X_{n}} U_{n},
S =^{def} X ×_{[X/G]}X = G ×_{Spec k} X,
S_{n} =^{def} X_{n} ×_{[X_{n}/G]}X_{n} = G ×_{Spec k} X_{n}.

The compatible isomorphisms ϕ_{n}: X_{n} ≃ U_{n} and X_{n} ≃ [X_{n}/G] yield yield isomorphisms of schemes in groupoids of R_{n} ⇉ U_{n} with S_{n} ⇉ X_{n}, for each n ≥ 0.

Denote by I_{U} the sheaf of ideals of the inverse image of Y × {0} ⊆ X × ^{1} in U × ^{1}, and by I_{R} the sheaf of ideals of its inverse image in R × ^{1}. Also denote with J_{U} the sheaf of ideals of Y × {0} ⊆ X × {0} in U × {0}, pushed forward to U × ^{1}, and by J_{R} the sheaf of ideals of its inverse image in R × {0}, pushed forward to R × ^{1}. There are natural surjection I_{U} ↠ J_{U} and I_{R} ↠ J_{R}. Set

U′ =^{def} Proj_{U× ^{1}} ( ⨁_{m=0}^{∞} I_{U}^{m}), R′ =^{def} Proj_{R× ^{1}} ( ⨁_{m=0}^{∞} I_{R}^{m}),

U′′=^{def} Proj_{U× ^{1}} ( ⨁_{m=0}^{∞} J_{U}^{m}), R′′=^{def} Proj_{R× ^{1}} ( ⨁_{m=0}^{∞} J_{R}^{m}).

Then R′⇉ U′is a scheme in groupoids, R′′⇉ U′′is a closed subgroupoid, and the difference groupoid R′∖ R′′⇉ U′∖ U′′gives a smooth presentation of M. Let us denote by U′_{n} and U′′_{n}the inverse images of U_{n} in U′and U′′, and by R′_{n} and R′′_{n} the inverse images of U_{n} × U_{n} in R′and R′′. Then the groupoid R′_{n} ∖ R′′_{n}⇉ U′_{n}∖ U′′_{n} gives a smooth presentation of M_{n}; furthermore, we have

U′_{n} = Proj_{U_{n}× ^{1}} ( ⨁_{m=0}^{∞} I_{U}^{m} ⊗_{O_{U× ^{1}}} O_{U_{n}× ^{1}} ) ,

R′_{n} = Proj_{R_{n}× ^{1}} ( ⨁_{m=0}^{∞} I_{R}^{m} ⊗_{O_{U× ^{1}}} O_{R_{n}× ^{1}} ) ,

U′′_{n}= Proj_{U_{n}× ^{1}} ( ⨁_{m=0}^{∞} J_{U}^{m} ⊗_{O_{R× ^{1}}} O_{U_{n}× ^{1}} ) ,

R′′_{n}= Proj_{R_{n}× ^{1}} ( ⨁_{m=0}^{∞} J_{R}^{m} ⊗_{O_{R× ^{1}}} O_{R_{n}× ^{1}} ).

In a completely analogous manner, denote by I_{X} the sheaf of ideals of Y × {0} in U × ^{1}, and by I_{S} the sheaf of ideals of its inverse image in S × ^{1}. Also denote with J_{X} the sheaf of ideals of Y × {0} in X × {0}, pushed forward to X × ^{1}, and by J_{S} the sheaf of ideals of its inverse image in S × {0}, pushed forward to S × ^{1}. Set

X′_{n} =^{def} Proj_{X_{n}× ^{1}} ( ⨁_{m=0}^{∞} I_{X}^{m} ⊗_{O_{X× ^{1}}} O_{X_{n}× ^{1}} ) ,

S′_{n} =^{def} Proj_{S_{n}× ^{1}} ( ⨁_{m=0}^{∞} I_{S}^{m} ⊗_{O_{S× ^{1}}} O_{S_{n}× ^{1}} ) ,

X′′_{n}=^{def} Proj_{U× ^{1}} ( ⨁_{m=0}^{∞} J_{U}^{m} ⊗_{O_{R× ^{1}}} O_{R_{n}× ^{1}} ) ,

S′′_{n} =^{def} Proj_{R× ^{1}} ( ⨁_{m=0}^{∞} J_{R}^{m} ⊗_{O_{R× ^{1}}} O_{R_{n}× ^{1}} ).

By the same argument as before, we see that [M_{n}/G] has a smooth presentation S′_{n} ∖ S′′_{n}⇉ X′_{n} ∖ X′′_{n}; hence, to complete the proof we need to establish the existence of isomorphisms U′_{n} ≃ X′_{n} and R′_{n} ≃ S′_{n}, compatible with the groupoid structures, the isomorphisms ϕ_{n}: U_{n} ≃ X_{n} and ψ_{n}: R_{n} ≃ S_{n}, and the embeddings ?_{n−1} ⊆?_{n}.

Let us denote by R̃ and Ũ the formal schemes obtained by completing R × 1 and U × ^{1} respectively along the inverse images of u_{0} ∈ U, and by S̃ and X̃ the formal schemes obtained by completing S × ^{1} and X × ^{1} respectively along the inverse images of the origin in X. The structure maps of the schemes in groupoids R_{n} ⇉ U_{n} and S_{n} ⇉ X_{n} pass to the limit, yielding formal schemes in groupoids R̃ ⇉ Ũ and S̃ ⇉ X̃. The isomorphisms ϕ_{n}: U_{n} ≃ X_{n} and ψ_{n}: R_{n} ≃ S_{n} give isomorphisms of formal schemes ϕ̃: Ũ ≃ X̃ and ψ̃: R̃ ≃ S̃, yielding an isomorphism of formal schemes in groupoids of R̃⇉ Ũ with S̃ ⇉ X̃.

Denote by IŨ and IR̃ the sheaves of ideals of the inverse images of Y × {0} ⊆ X × ^{1} in Ũ and R̃ respectively, and by JŨ and JR̃ the pushforwards to U and R̃ of the sheaves of ideals of the pullbacks of the inverse images of Y × {0} in the inverse images of X × {0}. Analogously, denote by IX̃ and IS̃ the sheaves of ideals of the inverse images of [Y/G] × {0} ⊆ [X/G] × ^{1} in X̃ and S̃ respectively, and by JX̃ and JS̃ the pushforwards to X̃ and S̃ of the sheaves of ideals of the pullbacks of the inverse images of [Y/G] × {0} in the inverse images of [X/G] × {0}.

The natural morphisms ũ: Ũ → U × ^{1}, r̃: R̃ → R × ^{1}, x̃: X̃ → X × ^{1} and s̃: S̃ → S × ^{1} are flat (this is actually the key point of this part of the proof). Furthermore, the inverse images of Y × {0} ⊆ X × ^{1} and of Y × {0} ⊆ X × ^{1} in Ũ and R̃ and the inverse images of [Y/G] × {0} ⊆ [X/G] × ^{1} and of [X/G] × {0} ⊆ [X/G] × ^{1} in Ũ and R̃ correspond under ϕ̃ and ψ̃; hence for each m ≥ 0 we obtain canonical isomorphisms of coherent sheaves

ũ^{∗}I_{U}^{m} ≃ ϕ̃^{∗} x̃^{∗}I_{X}^{m}, r̃^{∗}I_{R}^{m} ≃ ψ̃^{∗} S̃^{∗}I_{S}^{m},
ũ^{∗}J_{U}^{m} ≃ ϕ̃^{∗} x̃^{∗}J_{X}^{m}, r̃^{∗}J_{R}^{m} ≃ ψ̃^{∗} S̃^{∗}J_{S}^{m}.

By restricting to U_{n} and R_{n} we obtain isomorphism of coherent sheaves

I_{U}^{m} ⊗_{O_{U× ^{1}}} O_{U_{n}× ^{1}} ≃ ϕ_{n}^{∗}(I_{X}^{m} ⊗_{O_{X× ^{1}}} O_{X_{n}× ^{1}} ),
I_{R}^{m} ⊗_{O_{R× ^{1}}} O_{R_{n}× ^{1}} ≃ ϕ_{n}^{∗}(I_{S}^{m} ⊗_{O_{X× ^{1}}} O_{S_{n}× ^{1}} ),
J_{U}^{m} ⊗_{O_{U× ^{1}}} O_{U_{n}× ^{1}} ≃ ϕ_{n}^{∗}(J_{X}^{m} ⊗_{O_{X× ^{1}}} O_{X_{n}× ^{1}} ),
J_{R}^{m} ⊗_{O_{R× ^{1}}} O_{R_{n}× ^{1}} ≃ ϕ_{n}^{∗}(J_{S}^{m} ⊗_{O_{X× ^{1}}} O_{S_{n}× ^{1}} ).

By summing up over all m we obtain isomorphism of the corresponding Rees algebras, which yield the desired isomorphisms U′_{n} ≃ X′_{n} and R′_{n} ≃ S′_{n}.

This ends the proof of Lemma 5.1, and of the Theorem. ♠

On the basis of examples, the following generalization of Theorem 1.2 seems plausible.

Conjecture 5.3. Let X be an amenable stack over k. Let L be an extension of k, and let ξ be an object of X (Spec L). Then ed_{k}ξ ≤ ged_{k}X + dim R_{u}(\underline{Aut}_{L} ξ).

Here R_{u}G denotes the unipotent radical of G, as in Section 2.1. Unfortunately, the approach used in this section breaks down in the more general setting of the above conjecture: if the stabilizer is not reductive, the slice theorem does not apply.

6. Essential dimension of GL_{n}-quotients

Suppose that G is a special affine algebraic group over k acting on a scheme X locally of finite type over k. For each field L/k we have an equivalence between [X/G](L) and the quotient category for the action of the discrete group G(L) on the set X(L); hence the essential dimension of [X/G] equals the essential dimension of the functor of orbits

Orb_{G,X}: (Field/k) −→ (Set)

from the category (Field/k) of extensions of k to the category of sets, sending L to the set of orbits Orb_{G,X}(L) =^{def} X(L)/G(L); see [BRV09, Example 2.6].

For the rest of this section we will assume that X is an integral scheme, locally of finite type and smooth over k, and GL_{n} acts on X with generically finite stabilizers. Then the quotient stack [X/GL_{n}] is amenable; however the Genericity Theorem 1.2 does not tell us that ed_{k}[X/GL_{n}] = ged_{k}[X/GL_{n}] because we are not assuming that the stabilizer of every point of X is reductive. Nevertheless, in some cases one can still establish this equality by estimating ed_{k}ξ from above and proving, in an ad-hoc fashion, that ed_{k}(ξ) ≤ ged_{k}[X/GL_{n}] for every ξ ∈ [X/GL_{n}](L) whose automorphism group is not reductive. The rest of this section will be devoted to such estimates. These estimates will ultimately allow us to deduce Theorem 1.1 from Proposition 3.4.

For each positive integer λ, denote by J_{λ} the λ × λ Jordan block with eigenvalue 0, that is, the λ × λ matrix, that is, the linear transformation k^{λ} → k^{λ} defined by e_{1} ↦ 0 and e_{i} ↦ e_{i−1} for i = 2, …, λ, where e_{1}, …, e_{λ} is the canonical basis of k^{λ}. Let \underline{λ} be a partition of n; that is, \underline{λ} = (λ_{1}, …, λ_{r}) is a non-increasing sequence of positive integers with λ_{1} + ⋯+ λ_{r} = n. We denote by A\underline{λ} the n × n nilpotent matrix which is written in block form as

A\underline{λ} =^{def} ( \begin{matrix}J_{λ_{1}} & 0 & … & 0 \\ 0 & J_{λ_{2}} & … & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & J_{λ_{r}} \end{matrix} )

Every nilpotent n × n matrix is conjugate to a unique A\underline{λ}. Consider the 1-parameter subgroup ω\underline{λ} : _{a} → GL_{n} defined by ω\underline{λ}(t) = exp(tA\underline{λ}). We will usually assume that \underline{λ} ≠ (1^{n}); under this assumption ω\underline{λ} is injective. Denote by N\underline{λ} the normalizer of the image of ω\underline{λ}; the group N\underline{λ} acts on the fixed point locus X^{ω\underline{λ}} .

Lemma 6.1. Let L be a field extension of k, and ξ be an object of [X/GL_{n}](L) whose automorphism group scheme \underline{Aut}_{L} ξ is not reductive. Then

ed_{k}ξ ≤ max_{\underline{λ}} ed_{k}[X^{ω\underline{λ}} /N\underline{λ}] ,

where the maximum is taken over all partitions \underline{λ} of n different from (1^{n}).

Proof. Suppose ξ corresponds to the GL_{n}-orbit of a point p ∈ X(L). The automorphism group scheme \underline{Aut}_{L} ξ is isomorphic to the stabilizer G_{p} of p in GL_{n}. Since we are assuming that this group is not reductive, G_{p} will contain a copy of a, which is conjugate to the image of ω\underline{λ} for some \underline{λ} ≠ (1^{n}). After changing p to a suitable GL_{n}-translate, we may assume that the image of ω\underline{λ} is contained in G_{p}; hence p ∈ X^{ω\underline{λ}} . The composite X^{ω\underline{λ}} ↪ X → [X/GL_{n}] factors through [X^{ω\underline{λ}} /N\underline{λ}]; hence ξ is in the essential image of [X^{ω\underline{λ}} /N\underline{λ}](L) in [X/GL_{n}](L), and ed_{k}ξ ≤ ed_{k}[X^{ω\underline{λ}} /N\underline{λ}]. ♠

Lemma 6.2. ed_{k}[X^{ω\underline{λ}} /N\underline{λ}] ≤ dim X^{ω\underline{λ}} for any \underline{λ} ≠ (1^{n}).

Proof. By Lemma 2.1, N\underline{λ} is special. Hence,

ed_{k}[X^{ω\underline{λ}} /N\underline{λ}] = ed_{k}Orb_{N\underline{λ},X^{ω\underline{λ}}} ≤ dim X^{ω\underline{λ}} ,

as claimed. ♠

We now further specialize X to the affine space A_{n,d} of forms of degree d in the n variables \underline{x} = (x_{1}, …, x_{n}) over k. The general linear group GL_{n} acts on A_{n,d} in the usual way, via (Af)(\underline{x}) =^{def} f(\underline{x} · A^{−1} ) for any A ∈ GL_{n}. We are now ready for the main result of this section.

Theorem 6.3. Let L be a field extension of k, and ξ be an object of [A_{n,d}/GL_{n}](L) whose automorphism group scheme \underline{Aut}_{L} ξ is not reductive. Assume that either d ≥ 4 and n ≥ 2 or d = 3 and n ≥ 3. Then ed_{k}ξ ≤ \binom{n+d−1}{d} − n^{2}.

Proof. By Lemma 6.1 it suffices to show that

(6.1)  ed_{k}[X^{ω\underline{λ}} /N\underline{λ}] ≤ \binom{n + d − 1}{d} − n^{2} .

for any \underline{λ} ≠ (1^{n}). The space A_{n,d}^{ω\underline{λ}} consists of the forms f(\underline{x}) such that

f(exp(−tA\underline{λ}x)) = f(\underline{x}) .

By differentiating and applying the chain rule, this is equivalent to

(6.2)  ▽f(\underline{x}) · A\underline{λ} = 0 ,

where ▽f = (∂f/∂x_{1}, …, ∂f/∂x_{n}) is the gradient of f. We now proceed with the proof of (6.1)in three steps.

Case 1: Assume d ≥ 4. By Lemma 6.2 it suffices to show that

(6.3)  dim A_{n,d}^{ω\underline{λ}} ≤ \binom{n + d − 1}{d} − n^{2} .

for any \underline{λ} ≠ (1^{n}). For \underline{λ} ≠ (1^{n}) formula (6.2)tells us that ∂f/∂x_{1}, is identically zero. In other words, f(\underline{x}) is a form in x_{2}, …, x_{n}. Such forms lie in an affine subspace of A_{n,d} isomorphic to A_{n−1,d} . Hence,

dim A_{n,d}^{ω\underline{λ}} ≤ dim A_{n−1,d} = \binom{n + d − 2}{d} ,

and it suffices to prove the inequality

\binom{n + d − 1}{d} − \binom{n + d − 2}{d} ≥ n^{2}

or equivalently,

(6.4)  \binom{n + d − 2}{d − 1} ≥ n^{2} .

Since \binom{n+d−2}{d−1} = \binom{n+d−2}{n−1} is an increasing function of d for any given n ≥ 1, it suffices to prove (6.4)for d = 4. In this case

\binom{n + d − 2}{d − 1} − n^{2} = \binom{n + 2}{3} − n^{2} = \frac{n(n − 1)(n − 2)}{6} ≥ 0

for any n ≥ 2, as desired.

Case 2: d = 3 and \underline{λ} ≠ (1^{n}) or (2, 1^{n−1} ). Once again, it suffices to prove (6.3). If \underline{λ} ≠ (1^{n}) or (2, 1^{n−1} ) then (6.2)shows that for every f(\underline{x}) in A_{n,d}^{ω\underline{λ}} at least two of the partial derivatives ∂f/∂x_{1}, and ∂f/∂x_{i} are identically zero. For notational simplicity we will assume that i = 2. Then f(\underline{x}) is a form in the variables x_{3}, …, x_{n}. Hence,

dim A_{n,3}^{ω\underline{λ}} ≤ dim A_{n−2,3} = \binom{n}{3} = \binom{n + 2}{3} − n^{2} ,

as desired.

Case 3: Finally assume d = 3 and \underline{λ} = (2, 1^{n−1} ). Set ω =^{def} ω_{(2,1^{n−1} )} and N =^{def} N_{(2,1^{n−1} )}. By Lemma 2.1(b) N is a special group. Hence, we may identify the set of isomorphism classes in [A_{n,3}^{ω}/N](K) with the set of N(K)-orbits in [A_{n,3}^{ω}](K), for every field K/k.

By (6.2)A_{n,d}^{ω} consists of degree d forms f(x_{1}, …, x_{n}) such that ∂f/∂x_{1} = 0. That is, f(x_{1}, …, x_{n}) ∈ A_{n,d}^{ω} if and only if f is a form in the variables x_{2}, …, x_{n}. Thus A_{n,d}^{ω} is an affine subspace of A_{n,d} isomorphic to A_{n−1,d} . Our goal is to show that

(6.5)  ed_{k}f ≤ \binom{n + 2}{3} − n^{2}

for any f(\underline{x}) ∈ A_{n,d}^{ω}(K).

The normalizer N contains a subgroup

Γ ≃ GL_{n−2} ⋉ _{a}^{n−2}

consisting of matrices of the form

( \begin{matrix}1 & 0 & 0 & … & 0 \\ 0 & 1 & 0 & … & 0 \\ 0 & a_{3} \\ ⋮ & ⋮ & & A \\ 0 & a_{n} \end{matrix})

where (a_{3}, …, a_{n}) ∈ _{a}^{n−2} and A ∈ GL_{n−2} . We may assume without loss of generality that the stabilizer of f in Γ does not contain a non-trivial unipotent subgroup. Indeed, if it does then f is a Γ-translate of an element of A_{n,3}^{ω\underline{λ}} for some \underline{λ} ≠ (1^{n}) or (2, 1^{n−1} ). For such f the inequality (6.5)was established in Case 2.

Since both Γ and N are special, it is obvious that the essential dimension of f, viewed as an element of [A_{n,3}^{ω}/N](K) = Orb_{N,A_{n,3}^{ω}}(K) is no greater than the essential dimension of f, viewed as an element of [A_{n,3}^{ω}/Γ](K) = Orb_{Γ,A_{n,3}^{ω}}(K). Since we are assuming that the stabilizer of f in Γ does not contain any non-trivial unipotent subgroups, the Genericity Theorem 1.2 tells us that

ed_{k}f ≤ ged_{k}[A_{n,3}^{ω}/Γ] .

By Lemma 6.4 below for n ≥ 4 the action of Γ ⊂ GL_{n−1} on the space A_{n−1,3} of forms of degree 3 in the n − 1 indeterminates x_{2}, …, x_{n} is generically free. Thus [A_{n,3}/Γ] is an amenable stack and is generically a scheme. Consequently,

ged_{k}[A_{n,3}^{ω}/Γ_{n−2} ] = dim A_{n−1,3} − dim Γ_{n−2} = \binom{n + 1}{3} − (n − 2)^{2} − (n − 2) .

A simple computation shows that

\binom{n + 1}{3} − (n − 2)^{2} − (n − 2) ≤ \binom{n + 2}{3} − n^{2} ,

for any n ≥ 4; indeed,

\binom{n + 2}{3} − n^{2} − ( \binom{n + 1}{3} − (n − 2)^{2} − (n − 2))  = \frac{(n − 2)(n − 3)}{2} − 1 ≥ 0.  ♠

Lemma 6.4.

(a) Assume that the base field k is algebraically closed and G is a connected linear algebraic k-group such that N ∩ Z(G) ≠ {1} for every closed normal subgroup {1} ≠ N ◁ G. Here Z(G) denotes the center of G. (For example, G could be almost simple or GL_{n}.) Let H_{1}, H_{2} be closed subgroups of G such that H_{1} is finite and H_{2} contains no nontrivial central elements of G. Then for g ∈ G(k) in general position, H_{1} ∩ gH_{2}g^{−1} = {1}.

(b) Assume d ≥ 3 and n ≥ 1. Let GL_{n−1} be the subgroup of GL_{n} acting on the variables x_{2}, …, x_{n}. Then for f ∈ A_{n,d} in general position, Stab_{GL_{n−1}} (f) = {1}.

Proof. (a) Assume the contrary. Consider the natural (translation) action of H_{1} on the homogeneous space G/H_{2}. By our assumption this action is not generically free. Since H_{1} is finite, we conclude that this action is not faithful, i.e., some 1 ≠ h ∈ H_{1} acts trivially on G/H_{2}. Then h lies in N = ⋂ _{g∈G} gH_{2}g^{−1} . Consequently, N is a non-trivial normal subgroup of G. By our assumption N (and hence, H_{2}) contains a non-trivial central element of G, a contradiction.

(b) We may assume that k is algebraically closed. By [Ric72, Theorem A], there exists a subgroup S_{n,d} ⊂ GL_{n} and a dense open subset U ⊂ A_{n,d} such that Stab_{GL_{n}}(f) is conjugate to S for every f ∈ U. Moreover, for d ≥ 3 (and any n ≥ 1) S_{n,d} is a finite group; see Example 3.2.

Write f(x_{1}, …, x_{n}) = ∑ _{i=0}^{d} x_{1}^{d−i} f_{i}(x_{2}, …, x_{n}), where f_{i} is a form of degree i in x_{2}, …, x_{n}. Clearly g ∈ GL_{n−1} stabilizes f if and only if it stabilizes f_{1}, f_{2}, …, f_{d}. In other words,

Stab_{GL_{n−1}} (f) = ⋂_{i=1}^{d} Stab_{GL_{n−1}} (f_{i}) .

Moreover, each Stab_{GL_{n−1}} (f_{i}) is a conjugate of S_{n−1,i} in GL_{n−1} . Thus it suffices to show that for g_{1}, …, g_{d} in general position in GL_{n−1} ,

(6.6)  g_{1}S_{n−1,1} g_{1}^{−1} ∩ ⋯∩ g_{d}S_{n−1,d} g_{d}^{−1} = {1}.

This is a consequence of part (a), with G = GL_{n}, H_{1} = S_{n−1,1} and H_{2} = S_{n−1,d} . ♠

Remark 6.5. We note that if d ≥ 4 and n ≥ 3 or d = 3 and n ≥ 5 then (6.6)is immediate, since S_{n−1,d} = {1}; see Example (3.2). However, this argument does not cover the case where d = 3 and n = 3 or 4, which are needed for the proof of Theorem 6.3 above.

7. Proof of Theorem 1.1

Theorem 1.1(a) is an immediate consequence of what we have done so far. Indeed, by Proposition 3.4(a),

ged_{k}[A_{n,d}/GL_{n}] = \binom{n + d − 1}{d} − n^{2} + 1 + cd(GL_{n}/μ_{d}) .

Thus it suffices to show that for any field extension K/k and any K-point ζ of [A_{n,d}/GL_{n}](K), we have

ed_{k}ζ ≤ ged_{k}[A_{n,d}/GL_{n}] .

If the automorphism group scheme \underline{Aut}_{K}(ζ) is reductive, this is a direct consequence of Theorem 1.2, and if \underline{Aut}_{K}(ζ) is not reductive, then Theorem 6.3 tells us that

ed_{k}ζ ≤ \binom{n + d − 1}{d} − n^{2} < ged_{k}[A_{n,d}/GL_{n}] .

This completes the proof of Theorem 1.1(a). The rest of this section will be devoted to proving Theorem 1.1(b). The main complication here is that the stack [ (A_{n,d})/GL_{n}] is not amenable (see Example 3.2) and thus our Genericity Theorem 1.2 does not apply. We will get around this difficulty by relating [ (A_{n,d})/GL_{n}] to the amenable stack [ (A_{n,d})/PGL_{n}].

Proposition 7.1. Let X = [ (A_{n,d})/GL_{n}] and \overline{X} =^{def} [ (A_{n,d})/PGL_{n}], with the natural projection ϕ: X → \overline{X}. Then for any extension L/k and any L-point ξ: Spec L → X,

ed_{k}ξ ≤ ed_{k}ϕ(ξ) + cd(GL_{n}/μ_{d}) .

Proof. Note that X is a gerbe banded by _{m} over \overline{X}.

By the definition of ed_{k}ϕ(ξ) there exists an intermediate field k ⊂ K ⊂ L such that ϕ(ξ) descends to Spec K and trdeg_{k}K = ed_{k}ϕ(ξ). Moreover, ξ: Spec L → X_{K} factors through a point ξ_{0}: Spec L → X_{K}, as in the diagram below.

\xymatrix{ & X_{K} \ar[r] \ar[d] & X \ar[d]^{ϕ} \\ \overline{ξ}: Spec L \ar@{-->}[ur]^{ξ_{0}} \ar[r] & Spec K \ar[r] & \overline{X},}

Note that X_{K} is a _{m}-gerbe over K. So ξ_{0} (and hence, ξ) descends to some intermediate subfield of K ⊂ K_{0} ⊂ L such that trdeg_{K}K_{0} ≤ ed_{k}(X_{K}) = cd(X_{K}), where the last equality is Proposition 2.3(a). Let η be the generic point of \overline{X}. We know that the Brauer class of X_{η} has index dividing n and exponent dividing d; see the proof of Proposition 3.4. By Lemma 2.4 the same is true of the Brauer class of X_{K}. Therefore, by Lemma 2.2(c), cd(X_{K}) ≤ cd(GL_{n}/μ_{d}). In summary,

ed_{k}ξ ≤ trdeg_{k}K_{0} = trdeg_{k}K + trdeg_{K}K_{0} ≤ ed_{k}(ϕ(ξ)) + cd(X_{K}),

as claimed. ♠

Proof of Theorem 1.1(b).Let X =^{def} [ (A_{n,d})/GL_{n}], \overline{X} =^{def} [ (A_{n,d})/PGL_{n}], and Y =^{def} [A_{n,d}/GL_{n}] and consider the following diagram of natural maps.

\xymatrix{ & Y \ar[dl] \ar[ddl]^{ψ, a μ_{d}-gerbe} \\ X \ar[d]_{ϕ, a _{m}-gerbe} \\ \overline{X}}

In view of Proposition 7.1, it suffices to show that for every field extension L/k and every L-point ξ: Spec L → \overline{X},

(7.1)  ed_{k}ϕ(ξ) ≤ \binom{n + d − 1}{d} − n^{2} .

Recall from Example 3.2 that under our assumptions on n and d, we have that \overline{X} = [ (A_{n,d})/PGL_{n}] is amenable and is generically a scheme of dimension

dim (A_{n,d}) − dim PGL_{n} = \binom{n + d − 1}{d} − n^{2} .

If the automorphism group scheme \underline{Aut}_{L}(ϕ(ζ)) is reductive then (7.1)holds by the Genericity Theorem 1.2 applied to \overline{X}.

We may therefore assume that \underline{Aut}_{L}(ϕ(ζ)) is not reductive. Lift ξ to some ζ: Spec L → Y. (This can be done because Y → X is a _{m}-torsor.) Since the automorphism group scheme \underline{Aut}_{L}(ζ) is contained in the preimage of \underline{Aut}_{L}(ϕ(ζ)) under the natural projection map GL_{n} → PGL_{n}, we see that \underline{Aut}_{L}(ζ) is not reductive. Now ed_{k}ϕ(ξ) ≤ ed_{k}ζ and in view of Theorem 6.3

ed_{k}ζ ≤ \binom{n + d − 1}{d} − n^{2} .

This completes the proof of (7.1)and thus of Theorem 1.1(b). ♠

8. Small n and d

In this section we compute ed_{k}Forms_{n,d} and ed_{k}Hypersurf _{n,d} in the cases not covered by Theorem 1.1, building on the results of [BF03] and [BR05, Section 16].

To handle the case where n = 2, we need the following variant of [BR05, Lemma 16.1]. The proof is similar; we reproduce it here, with the necessary modifications, for the sake of completeness.

Lemma 8.1. ed_{k}Forms_{2,d} ≤ d − 1 and ed_{k}Hypersurf _{2,d} ≤ d − 2 for any d ≥ 3.

Proof. Let f(x_{1}, x_{2}) = a_{0}x_{1}^{d} + a_{1}x_{1}^{d−1} x_{2} + ⋯+ a_{d}x_{2}^{d} be a non-zero binary form of degree d over a field K/k. We claim that that f is equivalent (up to a linear coordinate changes by elements of GL_{2}(K)) to a binary form with (i) a_{0} = 0 or a_{1} = 0 and (ii) a_{d−1} = 0 or a_{d−1} = a_{d}. In each case f descends to the field k(a_{0}, …, a_{d}) and the hypersurface in ^{1} cut out by f descends to the field k(a_{i}/a_{j} |a_{j} ≠ 0). If (i) and (ii) are satisfied then the transcendence degrees of these fields over k are clearly ≤ d − 1 and d − 2, respectively. So, the lemma follows from the claim.

To prove the claim, we first reduce f to a form satisfying (i). If a_{0} = 0, we are done. If a_{0} ≠ 0, then performing the Tschirnhaus substitution

x_{1} ↦ x_{1} − \frac{a_{1}}{da_{0}}x_{2} , x_{2} ↦ x_{2}

we reduce f to a binary form with a_{1} = 0.

Now assume that f satisfies (i). We want to further reduce it to a form satisfying both (i) and (ii). If a_{d−1} = 0, we are done. If a_{d−1} ≠ 0, rescale x_{1} as follows

x_{1} ↦ \frac{a_{d}}{a_{d−1}} x_{1} , x_{2} ↦ x_{2} ,

to reduce f to a form satisfying (i) and a_{d−1} = a_{d}. This completes the proof of the claim and the lemma. ♠

Proposition 8.2. For any n ≥ 1 and d ≥ 2 we have

(a) ed_{k}Forms_{n,1} = ed_{k}Hypersurf _{n,1} = 0,

(b) ed_{k}Forms_{1,d} = 1 and ed_{k}Hypersurf _{1,d} = 0,

(c) ed_{k}Forms_{n,2} = n and ed_{k}Hypersurf _{n,2} = n − 1,

(d) ed_{k}Forms_{2,3} = 2 and ed_{k}Hypersurf _{2,3} = 1,

(e) ed_{k}Forms_{2,4} = 3 and ed_{k}Hypersurf _{2,4} = 2,

(f) ed_{k}Forms_{3,3} = 4 and ed_{k}Hypersurf _{3,3} = 3.

Proof. First we note that

(8.1)  ed_{k}Forms_{n,d} ≤ ed_{k}Hypersurf _{n,d}+1 .

This is easy to see directly from the definition or, alternatively, as a special case of the Fiber Dimension Theorem [BRV09, Theorem 3.2(b)], applied to the representable morphism of quotient stacks [A_{n,d} ∖ {0}/GL_{n}] → [ (A_{n,d})/GL_{n}] of relative dimension 1.

(a) Any non-zero linear form F(x_{1}, …, x_{n}) over any field K/k is equivalent to x_{1}.

(b) Degree d forms f_{1}(x) = ax^{d} and f_{2}(x) = bx^{d}, over a field K/k are equivalent if and only if b = ac^{d} for some c ∈ K^{∗} = GL_{1}(K). The assertions of part (b) follow easily from this.

(c) Any quadratic form F(x_{1}, …, x_{n}) over K/k can be diagonalized and hence, is defined over an intermediate field k ⊂ K_{0} ⊂ K such that trdeg_{k}K_{0} ≤ n. This implies that ed_{k}Forms_{n,2} ≤ n and ed_{k}Hypersurf _{n,2} ≤ n − 1. The opposite inequalities follow from [BR05, Proposition 16.2(b)].

(d) By Lemma 8.1, ed_{k}Forms_{2,3} ≤ 2 and ed_{k}Hypersurf _{2,3} ≤ 1, respectively. On the other hand, by [BR05, Proposition 16.2(c)], for the generic binary form F_{gen} in three variables (as in (1.3)) and the hypersurface H_{gen} it cuts out in ^{1}, we have ed_{k}F_{gen} = 2 and ed_{k}H_{gen} = 1.

Part (e) is proved in a similar manner, by combining Lemma 8.1 with [BR05, Proposition 16.2(d)].

(f) The identity ed_{k}Hypersurf _{3,3} = 3 is the main result of [BF03]. By (8.1), ed_{k}Forms_{3,3} ≤ 4.

In order to show that equality holds, it suffices to prove that the essential dimension ged_{k}[X_{3,3}/GL_{3}] of the generic form F_{gen} of degree 3 in 3 variables is at least 4. By [Ric72, Theorem A] the GL_{3}-action on X_{3,3} has a stabilizer in general position. Denote it by S_{3,3}, as in the proof of Lemma 6.4. As we mentioned there (and in Example 3.2), S_{3,3} is a finite subgroup of GL_{3}. Since the dimension of [X_{3,3}/GL_{3}] is 1, by [BR05, Lemma 15.4 and Proposition 5.5(c)] ed_{k}F_{gen} ≥ ed\overline{k}(S_{3,3}) + 1, where ed_{k}S_{3,3} denotes the essential dimension of the finite group S_{3,3}. (Note that in [BR05] the symbol ϕ_{n,d} was used in place of F_{gen}.) Thus it suffices to show that ed\overline{k}(S_{3,3}) ≥ 3.

To get a better idea about the structure of S_{3,3}, note that a form in X_{3,3}(\overline{k}) in general position is a scalar multiple of x_{1}^{3} + x_{2}^{3} + x_{3}^{3} + 3ax_{1}x_{2}x_{3} for some a ∈ \overline{k}. Hence, S contains a non-abelian subgroup H of order 27, generated by diagonal permutation matrices diag(ζ_{1}, ζ_{2}, ζ_{3}), where ζ_{1}, ζ_{2} and ζ_{3} are cube roots of unity satisfying ζ_{1}ζ_{2}ζ_{3} = 1, and the permutation matrices cyclically permuting x_{1}, x_{2} and x_{3}. Now

(8.2)  ed\overline{k}(S) ≥ ed\overline{k}(H) ≥ 3 ,

where the second inequality is a consequence of the Karpenko-Merkurjev theorem; see [MR10, Theorem 1.3]. This completes the proof of part (f). ♠

Remark 8.3.

(i) Since S is a finite subgroup of GL_{3}, it has a natural faithful 3-dimensional representation. Hence, ed_{k}S_{3,3} ≤ 3, and both inequalities in (8.2)are actually equalities.

(ii) The proof of part (e) shows that ed_{k}F_{gen} = 4, where F_{gen} is the generic form of degree 3 in 3 variables, as in (1.3). This answers an open question posed after the statement of Proposition 16.2 in [BR05].

9. Essential dimension of singular curves

In this section we use our new Genericity Theorem (1.2)to strengthen [BRV09, Theorem 7.3] on the essential dimension of the stack on (not necessarily smooth) local complete intersection curves with finite automorphism group presented in [BRV09, Theorem 7.3]. Let us recall the set-up. Denote by 𝔐_{g,n} the stack of all reduced n-pointed local complete intersection curves of genus g, that is, the algebraic stack over Spec k whose objects over a k-scheme T are finitely presented proper flat morphisms π: C → T, together with n sections s_{1}, …, s_{n}: T → C, where C is an algebraic space, the geometric fibers of π are connected reduced local complete intersection curves of genus g, and the image of each s_{i} is contained in the smooth locus of C → T. (We do not require the images of the sections to be disjoint.)

The stack 𝔐_{g,n} contains the stack M_{g,n} of smooth n-pointed curves of genus g as an open substack (here the sections are supposed to be disjoint). By standard results in deformation theory, every reduced local complete intersection curve is unobstructed, and is a limit of smooth curves. Furthermore there is no obstruction to extending the sections, since these map into the smooth locus. Therefore 𝔐_{g,n} is smooth and connected, and M_{g,n} is dense in 𝔐_{g,n}. However, the stack 𝔐_{g,n} is very large (it is certainly not of finite type), and in fact it is very easy to see that its essential dimension is infinite. Assume that we are in the stable range, i.e., 2g − 2 + n > 0: then in [BRV09] we show that the essential dimension of the open substack 𝔐_{g,n}^{fin} of 𝔐_{g,n} of curves with finite automorphism group equals the essential dimension of M_{g,n}.

Let C be an object of 𝔐_{g,n} defined over an algebraically closed field K. We say that C is reductive if the automorphism group scheme \underline{Aut}_{K}C is reductive. The marked curve C is not reductive if and only the smooth part C_{sm} ⊆ C contains a component that is isomorphic to _{K}^{1} and contains no marked points. A reductive object of 𝔐_{g,n} is an object C → S, whose geometric fibers over S are reductive. It is not hard to see that the reductive objects form an open substack 𝔐_{g,n}^{red} of 𝔐_{g,n}. Then our new genericity theorem applies, and allows to conclude that the essential dimensions of 𝔐_{g,n}^{red} and of M_{g,n} are the same. From [BRV09, Theorem 1.2] we obtain the following.

Theorem 9.1. If 2g − 2 + n > 0 and the characteristic of k is 0, then

ed_{k}𝔐_{g,n}^{red} = \{ \begin{matrix}2 & if (g, n) = (1, 1), \\ 5 & if (g, n) = (2, 0), \\ 3g − 3 + n & otherwise. \end{matrix}

References

[Alp09] Jarod Alper, On the local quotient structure of Artin stacks, arXiv:0904.2050, 2009.

[Art82] M. Artin, Brauer-Severi varieties, Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), Lecture Notes in Math., vol. 917, Springer, Berlin, 1982, pp. 194–210. MR 657430 (83j:14015)

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

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

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

[BRV09] P. Brosnan, Z. Reichstein, and A. Vistoli, Essential dimension of moduli of curves and other algebraic stacks, with an appendix by Najmuddin Fakhruddin, to appear in J. European Math. Society, 2009.

[CAS58] Chevalley C., Grothendieck. A., and J.-P. Serre, Séminaire C. Chevalley; 2e année: 1958. Anneaux de Chow et applications, Secrétariat mathématique, 11 rue Pierre Curie, Paris, 1958. MR 0110704 (22 #1572)

[CTKM07] J.-L. Colliot-Theélène, N. A. Karpenko, and A. S. Merkur′ev, Rational surfaces and the canonical dimension of the group PGL_{6}, Algebra i Analiz 19 (2007), no. 5, 159–178. MR MR2381945 (2008m:20074)

[Gab81] Ofer Gabber, Some theorems on Azumaya algebras, The Brauer group (Sem., Les Plans-sur-Bex, 1980), Lecture Notes in Math., vol. 844, Springer, Berlin, 1981, pp. 129–209.

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

[Jan04] Jens Carsten Jantzen, Nilpotent orbits in representation theory, Lie theory, Progr. Math., vol. 228, Birkhäuser Boston, Boston, MA, 2004, pp. 1–211. MR 2042689 (2005c:14055)

[KM97] Seán Keel and Shigefumi Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), no. 1, 193–213.

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

[LMB00a] Gérard Laumon and Laurent Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 39, Springer-Verlag, Berlin, 2000.

[McN06] George McNinch, On the centralizer of the sum of commuting nilpotent elements, J. Pure Appl. Algebra 206 (2006), no. 1-2, 123–140. MR 2220085 (2007a:17031)

[MM64] Hideyuki Matsumura and Paul Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1963/1964), 347–361. MR 0168559 (29 #5819)

[MR10] A. Meyer and Z. Reichstein, Some consequences of the Karpenko-Merkurjev theorem, Extra Volume: Andrei A. Suslin’s Sixtieth Birthday, Documenta Math., 2010, pp. 445–457.

[OS78] Peter Orlik and Louis Solomon, Singularities. II. Automorphisms of forms, Math. Ann. 231 (1977/78), no. 3, 229–240. MR 0476735 (57 #16292b)

[Ric72] R. W. Richardson, Jr., Principal orbit types for algebraic transformation spaces in characteristic zero, Invent. Math. 16 (1972), 6–14. MR 0294336 (45 #3405)

[San81] J.-J. Sansuc, Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres, J. Reine Angew. Math. 327 (1981), 12–80. MR 631309 (83d:12010)

[TV10] Dajano Tossici and Angelo Vistoli, On the essential dimension of infinitesimal group schemes, to appear in American J. Math. preprint available at http://arxiv.org/abs/1001.3988, 2010.

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

   E-mail address: reichst@math.ubc.ca

(Vistoli) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

   E-mail address: angelo.vistoli@sns.it