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We use a recent advance in birational geometry to prove new lower bounds on the essential dimension of some finite groups.

Suppose $G$ is a finite group and $p$ is either a prime number or $0$. For $p$ positive, we say that $G$ is weakly tame at $p$ if $G$ has no non-trivial normal $p$-subgroups. By convention we say that every finite group is weakly tame at $0$. Now suppose that $G$ is a finite group which is weakly tame at the residue characteristic of a discrete valuation ring $R$. Our main result shows that the essential dimension of $G$ over the fraction field $K$ of $R$ is at least as large as the essential dimension of $G$ over the residue field $k$. We also prove a more general statement of this type for a class of ��tale gerbes over $R$. As a corollary, we show that, if $G$ is weakly tame at $p$ and $k$ is any field of characteristic $p >0$ containing the algebraic closure of $\mathbb{F}_p$, then the essential dimension of $G$ over $k$ is less than or equal to the essential dimension of $G$ over any characteristic $0$ field. A conjecture of A. Ledet asserts that the essential dimension, $\mathrm{ed}_k(\mathbb{Z}/p^n\mathbb{Z})$, of the cyclic group of order $p^n$ over a field $k$ is equal to $n$ whenever $k$ is a field of characteristic $p$. We show that this conjecture implies that $\mathrm{ed}_{\mathbb{C}}(G) \geq n$ for any finite group $G$ which is weakly tame at $p$ and contains an element of order $p^n$. To the best of our knowledge, an unconditional proof of the last inequality is out of the reach of all presently known techniques., 16 pages. Corrected some minor mistakes, improved the exposition, and added some additional examples. To appear in Documenta Mathematica

We will calculate the essential dimension of the moduli stack of polarized K3 surfaces, including the positive char case., Some mistakes corrected

We determine the essential dimension of the spin group Spin(n) as an algebraic group over a field of characteristic 2, for n at least 15. In this range, the essential dimension is the same as in characteristic not 2. In particular, it is exponential in n. This is surprising in that the essential dimension of the orthogonal groups is smaller in characteristic 2. We also find the essential dimension of Spin(n) in characteristic 2 for n at most 10., Comment: 15 pages; v2: description of finite group scheme in proof of Theorem 2.1 corrected

We propose a generalization of Ledet’s conjecture, which predicts the essential dimension of cyclic p-groups in characteristic p, for finite commutative unipotent group schemes. And we present some evidence for this conjecture and discuss some consequences.

We prove that the essential dimension of central simple algebras of degree $p^{\ell m}$ and exponent $p^{m}$ over fields $F$ containing a base-field $k$ of characteristic $p$ is at least $\ell +1$ when $k$ is perfect. We do this by observing that the $p$-rank of $F$ bounds the symbol length in $\text{Br}_{p^{m}}(F)$ and that there exist indecomposable $p$-algebras of degree $p^{\ell m}$ and exponent $p^{m}$. We also prove that the symbol length of the Kato-Milne cohomology group $\text{H}_{p^{m}}^{n+1}(F)$ is bounded from above by $\binom{r}{n}$ where $r$ is the $p$-rank of the field, and provide upper and lower bounds for the essential dimension of Brauer classes of a given symbol length.

One of the important open problems in the theory of central simple algebras is to compute the essential dimension of GLn /μm, i.e., the essential dimension of a generic division algebra of degree n and exponent dividing m. In this paper we study the essential dimension of groups of the form G = (GLn1 × · · · ×GLnr )/C , where C is a central subgroup of GLn1 × · · · ×GLnr . Equivalently, we are interested in the essential dimension of a generic r-tuple (A1, . . . , Ar) of central simple algebras such that deg(Ai) = ni and the Brauer classes of A1, . . . , Ar satisfy a system of homogeneous linear equations in the Brauer group. The equations depend on the choice of C via the error-correcting code Code(C) which we naturally associate to C. We focus on the case where n1, . . . , nr are powers of the same prime. The upper and lower bounds on ed(G) we obtain are expressed in terms of coding-theoretic parameters of Code(C), such as its weight distribution. Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when r ≥ 3; in some cases we even compute the exact value. The Appendix by Athena Nguyen contains an explicit description of the Galois cohomology of groups of the form (GLn1 × · · · ×GLnr )/C. This description and its corollaries are used throughout the paper.

We characterize all fields of definition for a given coherent sheaf over a projective scheme in terms of projective modules over a finite-dimensional endomorphism algebra. This yields general results on the essential dimension of such sheaves. Applying them to vector bundles over a smooth projective curve C, we obtain an upper bound for the essential dimension of their moduli stack. The upper bound is sharp if the conjecture of Colliot-Th\'el\`ene, Karpenko and Merkurjev holds. We find that the genericity property proved for Deligne-Mumford stacks by Brosnan, Reichstein and Vistoli still holds for this Artin stack, unless the curve C is elliptic., Comment: 17 pages. v2: new counterexample to the genericity property in the elliptic curve case; the earlier Proposition 3.4 replaced by an argument which is still valid over non-perfect fields; exposition streamlined

Let $G$ be a permutation group on a set $\Omega$. A subset of $\Omega$ is a base for $G$ if its pointwise stabilizer is trivial; the base size of $G$ is the minimal cardinality of a base. In this paper we initiate the study of bases for algebraic groups defined over an algebraically closed field. In particular, we calculate the base size for all primitive actions of simple algebraic groups, obtaining the precise value in almost all cases. We also introduce and study two new base measures, which arise naturally in this setting. We give an application concerning the essential dimension of simple algebraic groups, and we establish several new results on base sizes for the corresponding finite groups of Lie type. The latter results are an important contribution to the classical study of bases for finite primitive permutation groups. We also indicate some connections with generic stabilizers for representations of simple algebraic groups., Comment: 62 pages; to appear in J. Eur. Math. Soc. (JEMS)

This paper shows that the number of independent parameters required to describe an Albert algebra up to isomorphism is at most seven. In other words, the essential dimension of the split group of type F 4 over a field of characteristic not 2 or 3 satisfies ed(F 4 )≤7 . This is achieved by reducing the structural group from the full 52-dimensional automorphism group to a subgroup of dimension 10, and exhibiting an eighteen-dimensional generically free linear representation that remains generically free once projectivized.

This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type E7 over a field of characteristic not 2 or 3. In particular, ed(E7) ≤ 29, and ed(E7, 2) ≤ 27.

The essential dimension is a numerical invariant of an algebraic group G which may be thought of as a measure of complexity of G-torsors over fields. A recent theorem of N. Karpenko and A. Merkurjev gives a simple formula for the essential dimension of a finite p-group. We obtain similar formulas for the essential p-dimension of a broad class of groups, which includes all algebraic tori.

We classify all finite groups of essential dimension 2 over an algebraically closed field of characteristic 0., Comment: 30 pages (To appear in Commentarii Mathematici Helvetici)

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 P^{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., Comment: 25 pages. Several typos in version 1 have been corrected

Essential dimension is an invariant of algebraic groups G over a field F that measures the complexity of G-torsors over field extensions of F. We use theorems of N. Karpenko about the incompressibility of Severi-Brauer varieties and quadratic Weil transfers of Severi-Brauer varieties to compute the essential dimension of some closed subgroups of RK/F (GL1(A)), where A is a central division K-algebra of prime power degree and K/F is a separable field extension of degree ≤ 2. In particular, we determine the essential dimension of the group Sim(A, σ) of similitudes of (A, σ), where σ is an F-involution on A, and the essential dimension of the normalizer \(N_{GL_1 (A)} \left( {GL_1 \left( B \right)} \right)\), where B is a separable subalgebra of A.

Let p be a prime integer, 1≤s≤r be integers and F be a field of characteristic different from p. We find upper and lower bounds for the essential p-dimension edp($ Al{{g}_{{{{p}^r},{{p}^s}}}} $) of the class $ Al{{g}_{{{{p}^r},{{p}^s}}}} $ of central simple algebras of degree pr and exponent dividing ps. In particular, we show that ed(Alg8,2)=ed2(Alg8,2)=8 and edp($ Al{{g}_{{{{p}^2},p}}} $)=p2+p for p odd.

We give upper bounds on the essential dimension of (quasi-)simple algebraic groups over an algebraically closed field that hold in all characteristics. The results depend on showing that certain representations are generically free. In particular, aside from the cases of spin and half-spin groups, we prove that the essential dimension of a simple algebraic group $G$ of rank at least two is at most $\mathrm{dim}(G) - 2(\mathrm{rank}(G)) - 1$. It is known that the essential dimension of spin and half-spin groups grows exponentially in the rank. In most cases, our bounds are as good or better than those known in characteristic zero and the proofs are shorter. We also compute the generic stabilizer of an adjoint group on its Lie algebra., v2 is a substantial revision

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

We prove that the essential dimension and p-dimension of a p-group G over a field F containing a primitive p-th root of unity is equal to the least dimension of a faithful representation of G over F.

LetKKbe an arbitrary field andGGa finite group. We will denote byedK⁡(G)\operatorname {ed}_K(G)the essential dimension ofGGoverKK. A generalization of the central extension theorem of Buhler and Reichstein (Compositio Math.106(1997) 159–179, Theorem 5.3) is obtained.

Let p be a prime number, let K be a field of characteristic not p, containing the p-th roots of unity, and let r≥1 be an integer. We compute the essential dimension of ℤ/p r ℤ over K (Theorem 4.1). In particular, i) We have edℚ(ℤ/8ℤ)=4, a result which was conjectured by Buhler and Reichstein in 1995 (unpublished). ii) We have edℚ(ℤ/p r ℤ)≥p r-1.

We generalize the result of Karpenko and Reichstein (2014) on incompressibility of Galois Weil transfer of generalized Severi–Brauer varieties to direct products of varieties of such type; as shown in Karpenko and Reichstein (2014), this is needed to compute essential dimension of representations of finite groups. We also provide a generalization to non-Galois (separable) Weil transfer.

We show that the essential dimension of a finite-dimensional central simple algebra coincides with the essential dimension of its r-linear trace form, $$ (a_1, \ldots, a_r) \mapsto tr(a_1, \ldots a_r), $$ for any r ≥ 3.

We define and study a new numerical invariant of an algebraic group action which we call the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in P^{n-1}., Comment: Significantly strenghtened, compared to the earlier version. To appear in Advances in Mathematics, 41 pages

We give an upper bound on the essential dimension of the group over the rational numbers, when q is a prime power.

Let k be a field and A be a separable k-algebra. Let r ≥ 3 be an integer. Generalizing a result of Reichstein (Arch. Math. 88 (2007), 12–18), we prove that the essential dimension over k of A equals that of the r-fold trace form $$F_r: (a_1,\ldots,a_r)\mapsto \mbox{tr}(a_1\ldots a_r)$$ .

We study the notion of essential dimension for a linear representation of a finite group. In characteristic zero we relate it to the canonical dimension of certain products of Weil transfers of generalized Severi-Brauer varieties. We then proceed to compute the canonical dimension of a broad class of varieties of this type, extending earlier results of the first author. As a consequence, we prove analogues of classical theorems of R. Brauer and O. Schilling about the Schur index, where the Schur index of a representation is replaced by its essential dimension. In the last section we show that essential dimension of representations can behave in rather unexpected ways in the modular setting., Comment: 24 pages. Corrected minor typos, added a section on essential dimension of modular representations

We study the essential dimension of exceptional connected simply connected algebraic groups over algebraically closed fields. In the present paper, we find upper estimates for the essential dimensions of the groupsF4,E6, andE7. For the groupF4, the upper estimate, thus obtained coincides with the known lower estimate. We also prove the second Serre conjecture for the groupE6 and for a function field.

The use of graphical and symbolic facilities in the teaching and learning of algebra and calculus will soon be a reality. Authors who write about the introduction of these instruments often claim that new technology is able to redress the imbalance between skill-dominated conceptions of school mathematics in favour of understanding. More recently some have stressed that `experimental mathematics' traditionally the reserve of mathematical research may be incorporated into the teaching and learning of mathematics. This paper looks into these two ideas and shows that they conceal an essential dimension: techniques play an important role in mathematical activity, intermediate between tasks and theories. This paper draws on research studies on the introduction of symbolic systems on computers and calculators and considers `new' techniques that accompany new technological instruments, their role in conceptualising and their links with `usual' paper/pencil techniques, as a key to analyse the role of technology in education. This view implies non obvious tasks for the teacher in the introduction of technology: the design of praxeologies adapted to new instrumental settings and everyday action on students' techniques.

We give a simple formula for the essential dimension of a finite pseudo-reflection group at a prime p and determine the absolute essential dimension for most irreducible pseudo-reflection groups. We also study the "poor man's essential dimension" of an arbitrary finite group, an intermediate notion between the absolute essential dimension and the essential dimension at a prime p., 24 pages. Final version. To appear in J. Lond. Math. Soc

Let $1\leq m \leq n$ be integers with $m|n$ and $\cat{Alg}_{n,m}$ the class of central simple algebras of degree $n$ and exponent dividing $m$. In this paper, we find new, improved upper bounds for the essential dimension and 2-dimension of $\cat{Alg}_{n,2}$. In particular, we show that $\ed_{2}(\cat{Alg}_{16,2})=24$ over a field $F$ of characteristic different from 2., Comment: Sections 1 and 3 are rewritten

Given an hermitian space we compute its essential dimension, Chow motive and prove its incompressibility in certain dimensions., Comment: 6 pages

Let K be an arbitrary field. We will determine explicitly all the nontrivial finite groups of essential dimension one over K., Comment: 26 pages

Two classes of surjective maps f: Sm -S' that are one-to-one over the image of a dense set are constructed. We show that for m, n > 3 there is a monotone surjection f: Sm -* Sn that is one-to-one over the image of a dense set; and for 3 < n < m < 2n 3, each element of 7rm(Sn) can be represented as a monotone surjection f: Sm -* Sn that is one-to-one over the image of a dense set.