The geometry of Ulrich bundles on del Pezzo surfaces

TL;DR

This paper characterizes when curves on del Pezzo surfaces correspond to the first Chern class of Ulrich bundles, linking geometric properties to the Minimal Resolution Conjecture and providing new examples of such curves.

Contribution

It establishes a criterion involving the kernel bundle for curves to represent the first Chern class of Ulrich bundles on del Pezzo surfaces, connecting to the Minimal Resolution Conjecture.

Findings

Curves on del Pezzo surfaces correspond to Ulrich bundle Chern classes if their kernel bundle admits a generalized theta-divisor.

On cubic surfaces, the existence of Ulrich bundles is equivalent to the Minimal Resolution Conjecture for certain curves.

New examples of complete intersection curves with semistable kernel bundles are constructed.

Abstract

Given a smooth del Pezzo surface $X_d \subseteq \mathbb{P}^{d}$ of degree $d,$ we show that a smooth irreducible curve $C$ on $X_d$ represents the first Chern class of an Ulrich bundle on $X_d$ if and only if its kernel bundle $M_C$ admits a generalized theta-divisor. This result is applied to produce new examples of complete intersection curves with semistable kernel bundle, and also combined with work of Farkas-Musta\c{t}\v{a}-Popa to relate the existence of Ulrich bundles on $X_d$ to the Minimal Resolution Conjecture for curves lying on $X_d.$ In particular, we show that a smooth irreducible curve $C$ of degree $3r$ lying on a smooth cubic surface $X_3$ represents the first Chern class of an Ulrich bundle on $X_3$ if and only if the Minimal Resolution Conjecture holds for $C.$