Counting $h^0(D)$ on primary Burniat surfaces

TL;DR

This paper develops an algorithm to compute cohomology of divisors on Burniat surfaces, demonstrating the non-existence of Ulrich line bundles and constructing a unique Ulrich rank 2 bundle, advancing understanding of vector bundles on these surfaces.

Contribution

It introduces a new algorithm for cohomology computation on Burniat surfaces and constructs a novel Ulrich rank 2 bundle not obtainable by previous methods.

Findings

No Ulrich line bundles exist on Burniat surfaces.

An Ulrich rank 2 vector bundle exists with respect to 3K_X.

The constructed Ulrich bundle cannot be derived from Casnati's method.

Abstract

We study the cohomology of divisors on a Burniat surface $X$ with $K_X^2=6$. We provide an algorithm for computing the cohomology groups of arbitrary divisors on $X$. As an application, we prove that there are no Ulrich line bundles\,(with respect to an arbitrary polarization), and that there exists an Ulrich vector bundle of rank 2 with respect to $3K_X$. The existence of Ulrich vector bundle of rank 2 was previously established by Casnati, but our construction yields one that cannot be obtained by his method.