On the positivity of the first Chern class of an Ulrich vector bundle

TL;DR

This paper investigates the positivity properties of the first Chern class of Ulrich vector bundles on smooth varieties, establishing conditions for bigness and classifying bundles with specific Chern class vanishing properties.

Contribution

It proves the positivity of the first Chern class on subvarieties not contained in lines and classifies certain Ulrich bundles with zero self-intersection of the first Chern class.

Findings

$c_1(E)$ is very positive outside lines in X

Ulrich bundles are big if X is not covered by lines

Classification of bundles with $c_1(E)^2=0$ on surfaces and threefolds

Abstract

We study the positivity of the first Chern class of a rank r Ulrich vector bundle E on a smooth n-dimensional variety $X \subseteq \mathbb P^N$. We prove that $c_1(E)$ is very positive on every subvariety not contained in the union of lines in X. In particular if X is not covered by lines, then E is big and $c_1(E)^n \ge r^n$. Moreover we classify rank r Ulrich vector bundles E with $c_1(E)^2=0$ on surfaces and with $c_1(E)^2=0$ or $c_1(E)^3=0$ on threefolds (with some exceptions).