Ulrich bundles on Del Pezzo threefolds

TL;DR

This paper studies the moduli space of Ulrich bundles on smooth Fano threefolds, proving smoothness, dimension, and existence conditions, revealing the wildness of these threefolds in the Ulrich context.

Contribution

It establishes the smoothness and dimension of the moduli space of Ulrich bundles on certain Fano threefolds and provides criteria for their existence based on special curves.

Findings

Moduli space of Ulrich bundles is smooth of dimension r^2+1 for index two threefolds.

No odd-rank Ulrich bundles exist on index four threefolds.

Any such threefold is Ulrich wild.

Abstract

We prove that for any $r \geq 2$ the moduli space of stable Ulrich bundles of rank $r$ and determinant $\mathcal O_X(r)$ on any smooth Fano threefold $X$ of index two is smooth of dimension $r^2+1$ and that the same holds true for even $r$ when the index is four, in which case no odd--rank Ulrich bundles exist. In particular this shows that any such threefold is Ulrich wild. As a preliminary result, we give necessary and sufficient conditions for the existence of Ulrich bundles on any smooth projective threefold in terms of the existence of a curve in the threefold enjoying special properties.