# Stability and deformations of generalised Picard sheaves

## TL;DR

This paper studies the stability and deformation properties of generalized Picard sheaves on moduli spaces of stable vector bundles over complex curves, leading to the construction of a fine moduli space for Picard bundle deformations.

## Contribution

It provides new results on the stability and deformation theory of generalized Picard sheaves, including conditions for local freeness and the construction of a moduli space for their deformations.

## Key findings

- Stability of generalized Picard sheaves is established under certain conditions.
- Deformations of locally free Picard sheaves are characterized and constructed.
- A fine moduli space for Picard bundle deformations is constructed when conditions are met.

## Abstract

Let $C$ be a smooth irreducible complex projective curve of genus $g \geq 2$ and $M$ the moduli space of stable vector bundles on $C$ of rank $n$ and degree $d$ with $\gcd(n,d)=1$. A generalised Picard sheaf is the direct image on $M$ of the tensor product of a universal bundle on $M\times C$ by the pullback of a vector bundle $E_0$ on $C$. In this paper, we investigate the stability of generalised Picard sheaves and, in the case where these are locally free, their deformations. When $g\ge3$, $n\ge2$ (with some additional restrictions for $g=3,4$) and the rank and degree of $E_0$ are coprime, this leads to the construction of a fine moduli space for deformations of Picard bundles.