Rank $2$ aCM and Ulrich bundles on Fano and Calabi--Yau double coverings of $\mathbb{P}^3$
Roberto Vacca
TL;DR
This work addresses the existence, classification, and geometry of arithmetically Cohen–Macaulay (aCM) and Ulrich sheaves on divisorial and cyclic coverings of projective space, with a focus on double covers of $\mathbb{P}^3$. It develops a general framework that translates aCM/Ulrich properties of sheaves on a covering $f:X\to\mathbb{P}^n$ into algebraic data given by square matrices, via a matrix factorization associated to the covering equation; the Hartshorne–Serre correspondence is used to study zero loci of rank 2 aCM bundles, yielding concrete geometric descriptions as complete intersections. For general double coverings of $\mathbb{P}^3$ branched in degrees $4,6,8$, the authors prove the existence of rank $2$ aCM and Ulrich bundles, and, when stable, determine the expected dimension of their moduli components; in the Calabi–Yau case ($m=4$), simple Ulrich bundles are spherical. The results extend Beauville–Beau determinant methods to divisorial coverings, provide a uniform description of rank $2$ aCM/Ulrich bundles and their zero loci, and offer moduli-theoretic consequences for Ulrich bundles on a broad family of varieties obtained as divisorial pullbacks from $\mathbb{P}^n$.
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.