On the essential dimension of symplectic vector bundles over curves

TL;DR

This paper calculates the essential dimension of the moduli stack of symplectic bundles over a curve, revealing a precise value linked to the generic gerbe's period, which differs from the vector bundle case.

Contribution

It provides the first exact computation of the essential dimension for symplectic bundles over curves, highlighting the role of the generic gerbe's period in this context.

Findings

Essential dimension of symplectic bundles is computed explicitly.

The generic gerbe of the moduli stack has period 2.

The result contrasts with the vector bundle case.

Abstract

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, we are able to precisely compute the essential dimension as the generic gerbe of the moduli stack has period 2 over it's moduli space.