Stable Ulrich Bundles on Fano Threefolds with Picard Number 2

TL;DR

This paper investigates the existence of stable Ulrich bundles of rank one and two on certain Fano threefolds with Picard number 2, identifying specific cases where they exist and constructing examples.

Contribution

It establishes the existence of stable rank two Ulrich bundles on a particular class of Fano threefolds obtained by blowing up a genus 3, degree 6 curve in projective space.

Findings

Ulrich line bundles exist only on the blow-up of a genus 3, degree 6 curve in P^3.

Stable rank two Ulrich bundles with c1=3H exist on a generic deformation of this class.

Abstract

In this paper, we consider the existence problem of rank one and two stable Ulrich bundles on imprimitive Fano 3-folds obtained by blowing-up one of $\mathbb{P}^{3}$, $Q$ (smooth quadric in $\mathbb{P}^{4}$), $V_{3}$ (smooth cubic in $\mathbb{P}^{4}$) or $V_{4}$ (complete intersection of two quadrics in $\mathbb{P}^{5}$) along a smooth irreducible curve. We prove that the only class which admits Ulrich line bundles is the one obtained by blowing up a genus 3, degree 6 curve in $\mathbb{P}^{3}$. Also, we prove that there exist stable rank two Ulrich bundles with $c_{1}=3H$ on a generic member of this deformation class.