Ulrich bundles on double coverings of projective space

TL;DR

This paper investigates the existence and properties of Ulrich bundles on double coverings of projective space, establishing conditions for their existence, constructing moduli spaces, and analyzing stability and restrictions.

Contribution

It demonstrates that certain double coverings admit rank 2 Ulrich sheaves, characterizes their zero loci, and constructs components of their moduli spaces with detailed stability analysis.

Findings

Double coverings of projective space admit Ulrich sheaves under specific conditions.

Rank 2 Ulrich sheaves exist for certain double coverings with n=3 and m=2,3,4.

Stable Ulrich bundles of all ranks are verified for m=2,3.

Abstract

Fixed a polarised variety $X$, we can ask if it admits Ulrich bundles and, in case, what is their minimal possible rank. In this thesis, after recalling general properties of Ulrich sheaves, we show that any finite covering of $\mathbb{P}^n$ that embeds as a divisor in a weighted projective space with weights $(1^{n+1},m)$ admits Ulrich sheaves, by using matrix factorisations. Among these varieties, we focus on double coverings of with $n\ge3$. Through Hartshorne--Serre correspondence, which we review along the way, we prove that the general such $X$ admits a rank $2$ Ulrich sheaf if and only if $n=3$ and $m=2,3,4$, and characterise the zero loci of their sections. Moreover, we construct generically smooth components of the expected dimension of their moduli spaces, analyse the action of the natural involution on them and the restriction of those bundles to low degree hypersurfaces. For $m=2,3$, we verify the existence of slope-stable Ulrich bundles of all the possible ranks.