Stability of projective Poincare and Picard bundles

TL;DR

This paper establishes the stability of projective Poincaré and Picard bundles over moduli spaces of stable vector bundles on algebraic curves, including cases where universal bundles do not exist, advancing understanding of their geometric properties.

Contribution

It proves the stability of projective Poincaré and Picard bundles on moduli spaces, even when universal bundles are absent, and extends stability results to bundles induced via reductive group homomorphisms.

Findings

Projective Poincaré bundle is stable with respect to any polarization.

Restriction of the projective Poincaré bundle to points is stable.

A projective Picard bundle is stable on certain open subsets of the moduli space.

Abstract

Let $X$ be an irreducible smooth projective curve of genus $g\ge3$ defined over the complex numbers and let ${\mathcal M}_\xi$ denote the moduli space of stable vector bundles on $X$ of rank $n$ and determinant $\xi$, where $\xi$ is a fixed line bundle of degree $d$. If $n$ and $d$ have a common divisor, there is no universal vector bundle on $X\times {\mathcal M}_\xi$. We prove that there is a projective bundle on $X\times {\mathcal M}_\xi$ with the property that its restriction to $X\times\{E\}$ is isomorphic to $P(E)$ for all $E\in\mathcal{M}_\xi$ and that this bundle (called the projective Poincar\'e bundle) is stable with respect to any polarization; moreover its restriction to $\{x\}\times\mathcal{M}_\xi$ is also stable for any $x\in X$. We prove also stability results for bundles induced from the projective Poincar\'e bundle by homomorphisms $\text{PGL}(n)\to H$ for any reductive $H$. We show further that there is a projective Picard bundle on a certain open subset $\mathcal{M}'$ of $\mathcal{M}_\xi$ for any $d>n(g-1)$ and that this bundle is also stable. We obtain new results on the stability of the Picard bundle even when $n$ and $d$ are coprime.