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

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).