Picard bundles and Twisted Picard bundles on the Jacobian of a curve

TL;DR

This paper investigates the properties of Picard and twisted Picard bundles on the Jacobian of a curve, revealing new stability results and embeddings into moduli spaces, with applications to stable ACM bundles and theta divisors.

Contribution

It introduces new stability and embedding results for Picard bundles on Jacobians of singular and smooth curves, expanding understanding of their geometric and moduli space properties.

Findings

Existence of a 2D family of stable ACM bundles with Picard bundles as limits for genus 2.

Construction of an embedding of the Jacobian into the moduli space of stable vector bundles.

Stability of the restriction of universal bundles and Picard bundles under certain conditions.

Abstract

Let $Y$ denote an irreducible projective curve with at most nodes as singularities and defined over an algebraically closed field of characteristic zero. We study the restriction of the twisted Picard bundles on the compactified Jacobian $\overline{J}(Y)$ of $Y$ to the embedded curve in $\overline{J}(Y)$. As an application, we show that for $g =2$ and each integer $r \ge 3$, there is a two-dimensional family of stable ACM bundles on the compactified Jacobian which has the Picard bundle in its limit. We define an embedding $\alpha_Y$ of the (generalised) Jacobian $J(Y)$ in the moduli space $U^s_Y(n,d)$ of stable vector bundles on $Y$ using a twisted restriction $E_Y$ of a Picard bundle to $Y \subset J(Y)$. We show that (under suitable conditions) the restriction of the universal bundle $\mathcal{U}$ to $Y \times J(Y)$ is stable for suitable polarisation. For the embedding of a smooth curve $Y$ given by $E_Y \otimes B, B$ a line bundle of degree $b$, we show that the restriction of the Picard bundle on $U^s_Y(n, d+nb)$ to $J(Y)$ is $\theta$-semistable for $b \ge 2g-1$ and $\theta$-stable for $b \ge 2g$. We also determine the relation between the restriction of the theta divisor on $U^s_Y(n,d+nb)$ to $J(Y)$ and the theta divisor $\theta$ on $\overline{J}(Y)$.