Ulrich bundles on rational surfaces with an anticanonical pencil

TL;DR

This paper proves the existence of stable rank 2 Ulrich bundles on certain rational surfaces with an anticanonical pencil, linking their cohomology, Chow forms, and Pfaffian representations.

Contribution

It demonstrates the existence of stable rank 2 Ulrich bundles on rational surfaces with an anticanonical pencil using Lazarsfeld-Mukai bundles, under mild assumptions.

Findings

Existence of stable rank 2 Ulrich bundles on these surfaces.

Chow forms are given by Pfaffians of skew-symmetric morphisms.

Ulrich bundles relate to the cohomology cone of projective varieties.

Abstract

Ulrich bundles are the simplest sheaves from the viewpoint of cohomology tables. Eisenbud and Schreyer conjectured that every projective variety carries an Ulrich bundle, which means it has the same cone of cohomology table as the projective space of same dimension. In this paper we show the existence of stable rank 2 Ulrich bundle on rational surfaces with an anticanonical pencil, under a mild Brill-Noether assumption by using Lazarsfeld-Mukai bundles. Also we see that each of those surfaces carries its Chow form given by the Pfaffian of skew-symmetric morphism coming from an Ulrich bundle.