Algorithms for singularities and real structures of weak Del Pezzo surfaces

TL;DR

This paper develops algorithms to classify singularities and real structures of weak Del Pezzo surfaces, extending existing classifications and enabling the construction of explicit examples.

Contribution

It introduces algorithms for classifying root subsystems related to singularities and real structures of weak Del Pezzo surfaces, providing a systematic computational approach.

Findings

Algorithms successfully classify singularities and real structures.

Constructed examples of weak Del Pezzo surfaces with specific singularities.

Extended classification to singular weak Del Pezzo surfaces.

Abstract

In this paper we consider the classification of singularities (Du Val) and real structures (Wall) of weak Del Pezzo surfaces from an algorithmic point of view. It is well known that the singularities of weak Del Pezzo surfaces correspond to root subsystems. We present an algorithm which computes the classification of these root subsystems. We represent equivalence classes of root subsystems by unique labels. These labels allow us to construct examples of weak Del Pezzo surfaces with the corresponding singularity configuration. Equivalence classes of real structures of weak Del Pezzo surfaces are also represented by root subsystems. We present an algorithm which computes the classification of real structures. This leads to an alternative proof of the known classification for Del Pezzo surfaces and extends this classification to singular weak Del Pezzo surfaces. As an application we classify families of real conics on cyclides.