Ulrich bundles on rational surfaces with an anticanonical pencil

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 implies it has the same cone of cohomology tables as the projective space of same dimension. In this paper we show the existence of stable rank 2 Ulrich bundles on rational surfaces with an anticanonical pencil, under a mild Brill-Noether assumption by using Lazarsfeld-Mukai bundles. As a consequence, the Chow form of a surface admitting a rank 2 Ulrich bundle of the form we construct is known to be the Pfaffian of a skew-symmetric matrix