A geometrical view of Ulrich vector bundles

TL;DR

This paper investigates the geometric properties of Ulrich vector bundles on smooth projective varieties, providing characterizations of ampleness and bigness, and classifying bundles with certain determinant properties using geometric maps.

Contribution

It offers new characterizations of ampleness and bigness of Ulrich bundles and their determinants, and classifies bundles with low numerical dimension of the determinant.

Findings

Characterization of ampleness of E and det E via restrictions to lines

Fibers of the map Φ_E are linear spaces

Classification of Ulrich bundles with low numerical dimension of det E

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 $\Phi_E :X \to {\mathbb G}(r-1, \mathbb PH^0(E))$ are linear spaces, as well as the projection on $X$ of all fibers of the map $\varphi_E : \mathbb P(E) \to \mathbb P H^0(E)$. 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 $\varphi_E$; 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 $\varphi_E$ 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.