Classification of singular del Pezzo surfaces over finite fields

TL;DR

This paper classifies singular del Pezzo surfaces over finite fields using Frobenius actions, extending existing classifications, and investigates which types can occur over specific finite fields, especially for degrees 3 to 6.

Contribution

It introduces an arithmetic classification of singular del Pezzo surfaces over finite fields and explores their possible types over given fields, extending prior work on ordinary del Pezzo surfaces.

Findings

Invariants depend only on the defined arithmetic type.

Certain types are shown to occur over specific finite fields.

Extension of classification to singular surfaces over finite fields.

Abstract

In this article, we consider weak del Pezzo surfaces defined over a finite field, and their associated, singular, anticanonical models. We first define arithmetic types for such surfaces, by considering the Frobenius actions on their Picard groups; this extends the classification of Swinnerton-Dyer and Manin for ordinary del Pezzo surfaces. We also show that some invariants of the surfaces only depend on the above type.Then we study an inverse Galois problem for singular del Pezzo surfaces having degree $3\leq d\leq 6$: we describe which types can occur over a given finite field (of odd characteristic when $3\leq d\leq 4$).