Related papers: Essential dimension of projective orthogonal and s…
We discuss the notion of essential dimension of a finite group and explain its relation with birational algebraic geometry. We show how this leads to a (partial) classification of simple finite groups of essential dimension less than or…
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…
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…
We prove a new upper bound on the essential p-dimension of the projective linear group PGLn.
The goal of this note is to reduce the existing upper bound for the essential dimension of central simple algebras of degree 8 and exponent 2 over fields of characteristic 2 from 10 to 9.
We provide a simple method to compute upper bounds on the essential dimension of split reductive groups with finite or connected center by means of their generically free representations. Combining our upper bound with previously known…
We define and study the essential dimension of an algebraic stack. We compute the essential dimension of the stacks Mgn and MgnBar of smooth, or stable, n-pointed curves of genus g. We also prove a general lower bound for the essential…
This is a survey of the existing literature, the state of the art, and a few minor new results and open questions regarding the essential dimension of central simple algebras and finite sequences of such algebras over fields whose…
Let $X$ be a smooth geometrically connected projective curve of genus at least 2 over a field of characteristic zero. We compute the essential dimension of the moduli stack of symplectic bundles over $X$. Unlike the case of vector bundles,…
To an orthogonal or unitary involution on a central simple algebra of degree 4, or to a symplectic involution on a central simple algebra of degree 8, we associate a Pfister form that characterises the decomposability of the algebra with…
In this paper, we compute the essential $p$-dimension of the split finite quasi-simple groups of classical Lie type at the defining prime, specifically the quasi-simple groups arising from the general linear and special linear groups, the…
The centers of the generic central simple algebras with involution are interesting objects in the theory of central simple algebras. These fields also arise as invariant fields for linear actions of projective orthogonal or symplectic…
One of the important open problems in the theory of central simple algebras is to compute the essential dimension of $\operatorname{GL}_n/\mu_m$, i.e., the essential dimension of a generic division algebra of degree $n$ and exponent…
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…
This article studies the set of R-equivalence classes of the group of proper projective similitudes of an algebra with involution of the first kind. The main results concern base fields of characteristic different from 2 over which every…
Let $p$ be a prime integer, $1\leq s\leq r$ integers, $F$ a field of characteristic $p$. Let $\cat{Dec}_{p^r}$ denote the class of the tensor product of $r$ $p$-symbols and $\cat{Alg}_{p^r,p^s}$ denote the class of central simple algebras…
We present the structure of symplectic cobordism ring $MSp_{*}$ in dimensions up to 51 and give a construction of an infinite series of elements $\Gamma_i$, $ i=1, 3,4, ...$, of order four in this ring, where $\operatorname{dim} \,…
We show that Nichols algebras of most simple Yetter-Drinfeld modules over the projective special linear group over a finite field, corresponding to non-semisimple orbits, have infinite dimension. We spell out a new criterium to show that a…
In this paper, we compute the essential $2$-dimension when the defining prime is odd of the general linear groups, the projective general linear groups, the special linear groups when $n$ is odd or $n = 2$, as well as the special linear…
We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or…