On the coregularity of del Pezzo surfaces with du Val singularities

TL;DR

This paper investigates the coregularity of del Pezzo surfaces with du Val singularities, linking their properties to elliptic fibrations and exploring differences over various fields, revealing most have zero coregularity.

Contribution

It provides a detailed computation of coregularity for del Pezzo surfaces with du Val singularities and establishes their relation to elliptic fibrations and field characteristics.

Findings

Most del Pezzo surfaces have coregularity 0.

Surfaces with positive coregularity relate to special isotrivial elliptic fibrations.

Results extend to non-algebraically closed fields in characteristic 0.

Abstract

We compute the coregularity of del Pezzo surfaces with du Val singularities. To this aim, we study the relation between del Pezzo surfaces of degree $1$ and elliptic fibrations. It turns out that del Pezzo surfaces with positive coregularity correspond to isotrivial elliptic fibrations with some special properties. We also prove results about coregularity of del Pezzo surfaces over non-algebraically closed fields of characteristic $0$. Our results confirm the expectation that "most" del Pezzo surfaces have coregularity $0$, while del Pezzo surfaces with positive coregularity enjoy some special properties.