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

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).

v2:Thm.1 and 7.2 include bigness;v3:added ampleness case in statement of Thm.1 and 7.2;v4:corrected a sign mistake in proof of Thm.7.2 (proves bigness for every n);v.5:removed one case in Thm.4 and modified the proof;v.6:corrected a few typos;v.7:changes thanks to referee:modified Def. 1.1 and 1.2, statement and proof of Thm.4, 5.1, 6.2, proof of 7.1 and of Thm.2. Thm.3, added 5.2, 7.3, 9.1

References in corpus (2)

• On Ulrich bundles on projective bundles
• On the existence of Ulrich bundles on blown-up varieties at a point