A geometrical view of Ulrich vector bundles

We study geometrical properties of an Ulrich vector bundle $E$ of rank $r$ on a smooth $n$-dimensional variety $X subseteq P^N$. 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 $Phi_{E}:X o {mathbb G}(r-1, P H^0(E))$ are linear spaces, as well as the projection on $X$ of all fibers of the map $arphi_{E} : P(E) o P H^0(E)$. 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 $arphi_{E}$; when $detE$ 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 $arphi_{E}$ have the same dimension; a classification of Ulrich vector bundles whose determinant has numerical dimension at most $rac{n}{2}$; a classification of Ulrich vector bundles with $det E$ of numerical dimension at most $k$ on a linear $P^k$-bundle