Genus 2 curves and generalized theta divisors

TL;DR

This paper studies generalized theta divisors on genus 2 curves, providing a desingularization and analyzing the theta map's behavior, revealing a linear embedding on general fibers.

Contribution

It introduces a desingularization of generalized theta divisors via a projective bundle and analyzes the theta map's restriction, uncovering new geometric properties.

Findings

Desingularization of theta divisors using a projective bundle.

The theta map restricts to a linear embedding on general fibers.

Enhanced understanding of vector bundles on genus 2 curves.

Abstract

In this paper we investigate generalized theta divisors $\Theta_r$ in the moduli spaces $\mathcal{U}_C(r,r)$ of semistable vector bundles on a curve $C$ of genus $2$. We provide a desingularization $\Phi$ of $\Theta_r$ in terms of a projective bundle $\pi:\mathbb{P}(\mathcal{V})\to\mathcal{U}_C(r-1,r)$ which parametrizes extensions of stable vector bundles on the base by $\mathcal{O}_C$. Then, we study the composition of $\Phi$ with the well known theta map $\theta$. We prove that, when it is restricted to the general fiber of $\pi$, we obtain a linear embedding.