Rank two aCM bundles on the del Pezzo fourfold of degree $6$ and its general hyperplane section

TL;DR

This paper classifies rank 2 aCM bundles on the del Pezzo fourfold of degree 6 and its hyperplane sections, extending known results for other del Pezzo varieties with specific Picard numbers.

Contribution

It provides a complete classification of rank 2 aCM bundles on a specific del Pezzo fourfold and its hyperplane sections, expanding the understanding of vector bundles on these varieties.

Findings

Complete classification of rank 2 aCM bundles on the del Pezzo fourfold of degree 6.

Extension of classification results to hyperplane sections of the fourfold.

Identification of conditions for the existence of such bundles.

Abstract

In the present paper we completely classify locally free sheaves of rank $2$ with vanishing intermediate cohomology modules on the image of the Segre embedding $\mathbb{P}^2\times\mathbb{P}^2\subseteq\mathbb{P}^8$ and its general hyperplane sections. Such a classification extends similar already known results regarding del Pezzo varieties with Picard numbers $1$ and $3$ and dimension at least $3$.