Related papers: Essential dimension and algebraic stacks
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 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…
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…
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 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…
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…
We find upper bounds on the essential dimension of the moduli stack of parabolic vector bundles over a curve. When there is no parabolic structure, we improve the known upper bound on the essential dimension of the usual moduli stack. Our…
We find upper bounds for the essential dimension of various moduli stacks of $\sln$-bundles over a curve. When $n$ is a prime power, our calculation computes the essential dimension of the stack of stable bundles exactly and the essential…
We determine the essential dimension of an arbitrary semisimple group of type $B$ of the form \[G=\big(\operatorname{\mathbf{Spin}}(2n_{1}+1)\times\cdots \times \operatorname{\mathbf{Spin}}(2n_{m}+1)\big)/\boldsymbol{\mu}\] over a field of…
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…
We study fundamental groups of algebraic stacks. We show that these fundamental groups carry an additional structure coming from the inertia groups. Then use this additional structure to analyze geometric/ topological properties of stacks.…
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…
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…
In this paper, we study the essential dimension of classes of central simple algebras with involutions of index less or equal to 4. Using structural theorems for simple algebras with involutions, we obtain the essential dimension of…
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…
We study the essential dimension of a finite group G over a field K. A generalization of the central extension theorem of Buhler and Reichstein (Compositio Math. 106 (1997) 159-179, Theorem 5.3) is obtained. We also get lower bounds of…
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…
We prove a new upper bound on the essential p-dimension of the projective linear group PGLn.
Consider the algebraic function $\Phi_{g,n}$ that assigns to a general $g$-dimensional abelian variety an $n$-torsion point. A question first posed by Kronecker and Klein asks: What is the minimal $d$ such that, after a rational change of…
Motivated by applications in moduli theory, we introduce a flexible and powerful language for expressing lower bounds on relative dimension of morphisms of schemes, and more generally of algebraic stacks. We show that the theory is robust…