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In this note we give simplified proofs of rectifiability of RCD(K,N) spaces as metric measure spaces and lower semicontinuity of the essential dimension, via -splitting maps. The arguments are inspired by the Cheeger-Colding theory for Ricci limits and rely on the second order differential calculus developed by Gigli and on the convergence and stability results by Ambrosio-Honda. peerReviewed

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

Essential dimension of a family of complex manifolds is the dimension of the image of its base in the Kuranishi space of the fiber. We prove that any family of hyperkahler manifolds over a compact simply connected base has essential dimension not greater than 1 . A similar result about families of complex tori is also obtained.

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.

Given a finite smooth group scheme $G$ over a field of characteristic $p > 0$, we show that the essential dimension of $G$ at $p$ is $0$ when $p$ does not divide the order of $G$, and $1$ when it does., 7 pages. Added a new proof of a key lemma due to a referee, and made some more minor changes. Accepted for publication in Comptes Rendus Mathematique

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

In this article the$p$-essential dimension of generic symbols over fields of characteristic$p$is studied. In particular, the$p$-essential dimension of the length$\ell$generic$p$-symbol of degree$n+1$is bounded below by$n+\ell$when the base field is algebraically closed of characteristic$p$. The proof uses new techniques for working with residues in Milne–Kato$p$-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on$p$-symbol algebras (i.e, degree 2 symbols) result from this work. The generic$p$-symbol algebra of length$\ell$is shown to have$p$-essential dimension equal to$\ell +1$as a$p$-torsion Brauer class. The second is a lower bound of$\ell +1$on the$p$-essential dimension of the functor$\operatorname{Alg}_{p^{\ell },p}$. Roughly speaking this says that you will need at least$\ell +1$independent parameters to be able to specify any given algebra of degree$p^{\ell }$and exponent$p$over a field of characteristic$p$and improves on the previously established lower bound of 3.

In the paper we survey research on the essential dimension. The highlights of the survey are the computations of the essential dimensions of finite groups, groups of multiplicative type and the spinor groups. We present self-contained proofs of these cases and give applications in the theory of simple algebras and quadratic forms.

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

We prove that spin groups act generically freely on various spinor modules, in the sense of group schemes and in a way that does not depend on the characteristic of the base field. As a consequence, we extend the surprising calculation of the essential dimension of spin groups and half-spin groups in characteristic zero by Brosnan--Reichstein--Vistoli (Annals of Math., 2010) and Chernousov--Merkurjev (Algebra & Number Theory, 2014) to fields of characteristic different from 2., v2 adds an appendix by Alexander Premet on generic stabilizer in HSpin_16 in characteristic 2

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 of the spinor group Spin_n grows exponentially with n; in particular, we give a precise formula for this essential dimension when n is not divisible by 4. We use this result to show that the number of 3-fold Pfister forms needed to represent the Witt class of a general quadratic form of rank n with trivial discriminant and Hasse-Witt invariant grows exponentially with n. This paper overlaps with our earlier preprint arXiv:math/0701903 . That preprint has splintered into several parts, which have since acquired a life of their own. In particular, see "Essential dimension of moduli of curves and other algebraic stacks", by the same authors, and "Some consequences of the Karpenko-Merkurjev theorem", by Meyer and Reichstein (arXiv:0811.2517)., Comment: 11 pages. Accepted for publication in Annals of Mathematics

In a previous paper the author determined all possible cohomological invariants of Aut ( J ) -torsors in Galois cohomology with mod 2 coefficients, where J is a split central simple Jordan algebra of odd degree n ⩾ 3 . In the present paper we extend these results to a larger class of groups G n 1 , … , n d r indexed by r = 0 , 1, 2 or 3, and n i positive integers that add to n ⩾ 3 such that n 1 is odd. We also give precise values for the essential dimension of these groups.

The present paper deals with algebraic tori and essential dimension in three unrelated contexts. After some preliminaries on essential dimension, versal torsors and tori, we explicitly construct a versal torsor for PGL n , n ⩾ 5 odd, defined over a field of transcendence degree 1 2 ( n − 1 ) ( n − 2 ) over the base field. This recovers a result of Lorenz, Reichstein, Rowen and Saltman. We also discuss the so-called “tori method” which gives a geometric proof of a result of Ledet on the essential dimension of a cyclic p-group. In the last section we compute the essential dimension of the functor K ↦ H 1 ( K , GL n ( Z ) ) , the latter set being in bijection with the isomorphism classes of n-dimensional K-tori.

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.

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.

In the present paper, we provide a lower bound of the essential dimension over a field of positive characteristic via Kato’s cohomology group, defined by cokernel of a general Artin–Schreier operator. Combining this with Tignol’s result on the second trace form of simple algebras of degree [Formula: see text], we show that [Formula: see text] over a field of characteristic [Formula: see text].

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.

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

We give an upper bound for the essential dimension of a smooth unipotent algebraic group over an arbitrary field. We also show that over a field $k$ which is finitely generated over a perfect field, a smooth unipotent algebraic $k$-group is of essential dimension 0 if and only if it is $k$-split., 24 pages

Following up on our earlier work and the work of N. Karpenko and A. Merkurjev, we study the essential p-dimension of linear algebraic groups G whose connected component G^0 is a torus., 23 pages, no figures. arXiv admin note: text overlap with arXiv:0910.5574

We affirmatively answer a conjecture in the preprint ``Essential dimension and algebraic stacks,'' proving that the essential dimension of an abelian variety over a number field is infinite., 4 pages. To appear in C. R. Math. Acad. Sci. Paris. Preprint posted earlier to http://www.mathematik.uni-bielefeld.de/LAG/

A conic bundle or quadric bundle in characteristic 2 can have generic fiber which is nowhere smooth over the function field of the base variety. In that case, the generic fiber is called a quasilinear quadric. We solve some of the main problems of birational geometry for quasilinear quadrics. First, if there is a rational map from one quasilinear quadric to another of the same dimension, and also a map back, then the two quadrics are birational. Next, we determine exactly which quasilinear quadrics are ruled, that is, birational over the base field to the product of some variety with the projective line. Both statements are conjectured for quadrics in any characteristic. The proofs begin by extending Karpenko and Merkurjev's theorem on the essential dimension of quadrics to arbitrary quadrics (smooth or not) in characteristic 2., 15 pages

Let $G$ be a finite group and $\phi\colon V\to W$ an equivariant morphism of finite dimensional $G$-modules. We say that $\phi$ is faithful if $G$ acts faithfully on $\phi(V)$. The covariant dimension of $G$ is the minimum of the dimension of $\bar{\phi(V)}$ taken over all faithful $\phi$. In this paper we investigate covariant dimension and are able to determine it for abelian groups and to obtain estimates for the symmetric and alternating groups. We also classify groups of covariant dimension less than or equal to 2. A byproduct of our investigations is the existence of a purely transcendental field of definition of degree $n-3$ for a generic field extension of degree $n\geq 5$., Comment: 22 pages

We give necessary and sufficient conditions for a finite group to have essential dimension 1 over an infinite ground field.