On the Unirationality of del Pezzo surfaces of degree two

TL;DR

This paper investigates the unirationality of degree two del Pezzo surfaces over various fields, extending previous work and proving most such surfaces over finite fields are unirational, with a few exceptions.

Contribution

It extends Manin's work on rational curves on del Pezzo surfaces and establishes the unirationality of almost all degree two del Pezzo surfaces over finite fields.

Findings

Most degree two del Pezzo surfaces over finite fields are unirational.

Identifies three explicit surfaces that may not be unirational.

Extends previous methods for finding rational curves on these surfaces.

Abstract

Among geometrically rational surfaces, del Pezzo surfaces of degree two over a field k containing at least one point are arguably the simplest that are not known to be unirational over k. Looking for k-rational curves on these surfaces, we extend some earlier work of Manin on this subject. We then focus on the case where k is a finite field, where we show that all except possibly three explicit del Pezzo surfaces of degree two are unirational over k.