On the essential dimension of coherent sheaves

Abstract

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-Thélène, 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.2000 Mathematics Subject Classification. 14D23, 14D20. Key words and phrases. Essential dimension, moduli stack, endomorphism algebra, curve. I. B. is supported by the J. C. Bose Fellowship. A. D. is partially supported by NSERC. N. H. was partially supported by SFB 647: Space -Time -Matter in Berlin. He thanks TIFR Bombay for hospitality, and Bernd Kreussler for a useful discussion on bundles over elliptic curves. Projective Modules over Right-Artinian RingsLet R be a ring. Our rings are always associative, and they always have a unit, but they are not necessarily commutative. By an R-module, we mean a right R-module, unless stated otherwise. Let n ⊂ R be a nilpotent two-sided ideal.Lemma 2.1. Every element q ∈ R/n with q 2 = q admits a lift p ∈ R with p 2 = p.Proof. By assumption, there is an integer n ≥ 1 such that n n = 0. Using induction over n, we may assume n 2 = 0 without loss of generality.

Essential dimension and genericity for quiver representations

“…As noted in the Introduction, part of the statement of Theorem 1.1 depends on a conjecture due to Colliot-Thélène Karpenko and Merkurjev, formulated in [8, §1]. Following [3], we rephrase this conjecture in a way that is better suited to our needs.…”

“…The genericity property fails in general (see [5,Example 6.5]) but holds for smooth algebraic stacks with reductive automorphism groups [23] (and in particular, Deligne-Mumford stacks [5]). In many interesting examples where these conditions are not satisfied, the genericity property continues to hold [3,23]. This phenomenon is poorly understood; one of the goals of this paper is to investigate the genericity property of stacks of quiver representations.…”

Incompressibility of Products of Pseudo-homogeneous Varieties

“…Corollary 9 shows that this assumption is superfluous. Another area of applications for this result is provided in [2], where it is important that the assumption of being balanced can be dropped.…”

The genericity theorem for the essential dimension of tame stacks

“…(1) plays a pivotal role in the calculation of the essential dimension of many stacks of geometric interest, such as stacks of smooth or stable curves, stacks of principally polarized abelian varieties [BRV11], coherent sheaves on smooth curves [BDH18], quiver representations [Sca20] and polarized K3 surfaces [Gao20].…”