On the essential dimension of coherent sheaves

TL;DR

This paper investigates the essential dimension of coherent sheaves on projective schemes, providing bounds for their moduli stacks and exploring the genericity property in relation to elliptic curves.

Contribution

It characterizes fields of definition for coherent sheaves via endomorphism algebras and establishes bounds for the essential dimension of their moduli stacks, extending known genericity results.

Findings

Upper bound for the essential dimension of vector bundle moduli stacks.

The bound is sharp under certain conjectures.

The genericity property extends to this setting except for elliptic curves.

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-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.