Projective Poincar\'{e} and Picard bundles for moduli spaces of vector bundles over nodal curves

TL;DR

This paper studies the stability of projective Poincaré and Picard bundles over moduli spaces of vector bundles on nodal curves, establishing their stability and describing the Picard group structure in certain cases.

Contribution

It proves the stability of these bundles over nodal curves and characterizes the Picard group for specific moduli spaces, extending known results to singular curves.

Findings

Projective Poincaré and Picard bundles are stable under suitable polarizations.

The restriction of the Poincaré bundle to a nonsingular point is stable.

The Picard group of the moduli space is isomorphic to Z for certain genus and degree conditions.

Abstract

Let $U^{'s}_L(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y \times U^{'s}_L(n,d)$ and the projective Picard bundle on $U^{'s}_L(n,d)$ are stable for suitable polarisation. For a nonsingular point $x \in Y$, we show that the restriction of the projective Poincare bundle to $x \times U^{'s}_L(n,d)$ is stable for any polarisation. We prove that for arithmetic genus $g\ge 3$ and for $g=n=2, d$ odd, the Picard group of the moduli space $U'_L(n,d)$ of semistable vector bundles of rank $n$ with determinant $L$ of degree $d$ is isomorphic to $\mathbb{Z}$.