A geometrical view of Ulrich vector bundles

Abstract

We study geometrical properties of an Ulrich vector bundle $E$ of rank $r$ on a smooth $n$-dimensional variety $X\subseteq {\mathbb{P}}^N$. We characterize ampleness of $E$ and of $\det E$ in terms of the restriction to lines contained in $X$. We prove that all fibers of the map ${\Phi }_E:X\to \mathbb{G}(r-1,\mathbb{P}{H}^0(E))$ are linear spaces, as well as the projection on $X$ of all fibers of the map ${\phi }_E:\mathbb{P}(E)\to \mathbb{P}{H}^0(E)$. Then we get a number of consequences: a characterization of bigness of $E$ and of $\det E$ in terms of the maps ${\Phi }_E$ and ${\phi }_E$; when $\det E$ is big and $E$ is not big there are infinitely many linear spaces in $X$ through any point of $X$; when $\det E$ is not big, the fibers of ${\Phi }_E$ and ${\phi }_E$ have the same dimension; a classification of Ulrich vector bundles whose determinant has numerical dimension at most $\frac{n}{2}$; a classification of Ulrich vector bundles with $\det E$ of numerical dimension at most $k$ on a linear ${\mathbb{P}}^k$-bundle.

v2: statement and proof of Thm.2 have been modified to allow scheme-theoretic fibers; Lemma 3.4 modified accordingly; added Rmk. 4.4