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

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.