Ulrich line bundles on double planes

TL;DR

This paper investigates the conditions under which Ulrich line bundles exist on double covers of the projective plane branched along smooth curves, providing classifications and examples for all degrees.

Contribution

It characterizes geometric conditions for Ulrich line bundles on double planes and classifies surfaces admitting them for all degrees.

Findings

Identifies geometric criteria for Ulrich line bundles on double planes.

Classifies surfaces with Ulrich line bundles for all degrees s ≥ 1.

Provides explicit examples of such surfaces.

Abstract

Consider a smooth complex surface $X$ which is a double cover of the projective plane $\mathbb{P}^2$ branched along a smooth curve of degree $2s$. In this article, we study the geometric conditions which are equivalent to the existence of Ulrich line bundles on $X$ with respect to this double covering. Also, for every $s\geq 1$, we describe the classes of such surfaces which admit Ulrich line bundles and give examples.