Instanton Sheaves on Ruled Fano 3-folds of Picard Rank 2 and Index 1

TL;DR

This paper investigates rank 2 instanton sheaves on certain Fano threefolds, showing their construction via elementary transformations and establishing the existence of instanton and Ulrich bundles on these varieties.

Contribution

It demonstrates that non-locally free instanton sheaves can be derived from locally free ones and proves the existence of instanton and Ulrich bundles on ruled Fano threefolds of Picard rank 2 and index 1.

Findings

Non-locally free instanton sheaves are elementary transformations of locally free sheaves.

Existence of orientable rank 2 instanton bundles on ruled Fano threefolds.

Existence of Ulrich bundles corresponding to minimal charge instanton sheaves.

Abstract

We study rank 2 $h$-instanton sheaves on projective threefolds. We demonstrate that any orientable rank 2, non-locally free $h$-instanton sheaf with defect 0 on a threefold can be obtained as an elementary transformation of a locally free $h$-instanton sheaf. Our focus then shifts to ruled Fano threefolds of Picard rank 2 and index 1, of which there are five deformation classes. We establish the existence of orientable rank 2 $h$-instanton bundles on such varieties. Additionally, we prove the existence of Ulrich bundles on such varieties, which correspond to $h$-instanton sheaves of minimum charge.