Full Degree Two del Pezzo Surfaces over Small Finite Fields

TL;DR

This paper extends the classification of split del Pezzo surfaces over finite fields, focusing on degree two surfaces where all points lie on the fifty-six exceptional curves, building on Hirschfeld's earlier work.

Contribution

It advances the classification of split degree two del Pezzo surfaces over finite fields by analyzing the distribution of points on exceptional curves.

Findings

Classification of split degree two del Pezzo surfaces over small finite fields.

Identification of conditions for points to lie on all exceptional curves.

Extension of Hirschfeld's classification to degree two surfaces.

Abstract

Hirschfeld classified split del Pezzo surfaces of degree at least three whose points are all contained on the lines in the surface. We continue his work and begin the classification of split degree two del Pezzo surfaces over finite fields whose points are all on the fifty-six exceptional curves of the surfaces.