A Geometrical View of Ulrich Vector Bundles

Abstract

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