Rank $2$ aCM and Ulrich bundles on Fano and Calabi--Yau double coverings of $\mathbb{P}^3$

TL;DR

This paper establishes the existence and classification of rank 2 aCM and Ulrich sheaves on special double coverings of projective space, analyzing their geometric properties and moduli space dimensions.

Contribution

It provides a comprehensive existence, classification, and geometric analysis of aCM and Ulrich sheaves on double coverings of projective space, including moduli space dimensions.

Findings

Existence of aCM and Ulrich sheaves on certain double coverings.

Classification of rank 2 sheaves on regular double coverings.

Dimension calculations for moduli spaces of stable sheaves.

Abstract

We prove existence of aCM and Ulrich sheaves respect to ample and globally generated polarisations on a class of special finite coverings $f:X\to\mathbb{P}^n$, which in particular contains cyclic ones. In the case of rank $2$ on double coverings, we have a precise description of the zero loci of such sheaves which allows us to study their geometry and classify all possible such bundles in the case $X$ is regular. We show that on a general double covering of $\mathbb{P}^3$ branched along a divisor of degree $4,6,8$ all the above sheaves exist and, when stable, we compute the dimension of their component in the moduli spaces.