# Theta divisors and Ulrich bundles on Geometrically ruled surfaces

## TL;DR

This paper explores the conditions under which geometrically ruled surfaces support Ulrich line bundles, revealing a surprising link with Theta divisors on moduli spaces and providing new existence results for Ulrich bundles.

## Contribution

It establishes a novel connection between Theta divisors on moduli spaces and Ulrich line bundles on ruled surfaces, leading to comprehensive existence results.

## Key findings

- Characterization of invariants for Ulrich line bundles on ruled surfaces
- Relation between Theta divisors and Ulrich bundle existence
- Existence of large families of Ulrich bundles in rank two case

## Abstract

We consider the following question: for which invariants $g$ and $e$ is there a geometrically ruled surface $S \rightarrow C$ over a curve $C$ of genus $g$ with invariant $e$ such that $S$ is the support of an Ulrich line bundle with respect to a very ample line bundle? A surprising relation between the existence of certain proper Theta divisors on some moduli spaces of vector bundles on $C$ with the existence of Ulrich line bundles on $S$ will be the key to completely solve the above question. The relation is realized by translating the vanishing conditions characterizing Ulrich line bundles to specific geometric conditions on the symmetric powers of the defining vector bundle of a given ruled surface. This general principle leads to some finer existence results of Ulrich line bundles in particular cases. Another focus is on the rank two case where, with very few exceptions, we show the existence of large families of special Ulrich bundles on arbitrary polarized ruled surfaces.