Regular del Pezzo surfaces with irregularity

TL;DR

This paper constructs the first known examples of regular del Pezzo surfaces with nonzero irregularity and establishes restrictions on their anti-canonical degree and irregularity.

Contribution

It provides the first explicit examples of regular del Pezzo surfaces with irregularity and determines possible pairs of anti-canonical degree and irregularity.

Findings

First examples of regular del Pezzo surfaces with irregularity

Restrictions on anti-canonical degree and irregularity pairs

New insights into the structure of del Pezzo surfaces

Abstract

We construct the first examples of regular del Pezzo surfaces for which the irregularity (i.e. the dimension of the first cohomology group of the structure sheaf) is nonzero. We also find a restriction on the integer pairs that are possible as the anti-canonical degree and irregularity of such a surface.