Unirationality of del Pezzo surfaces of degree two over finite fields

TL;DR

This paper proves that all degree two del Pezzo surfaces over finite fields are unirational, extending previous results and providing conditions for unirationality over more general fields.

Contribution

It establishes the unirationality of all degree two del Pezzo surfaces over finite fields and offers criteria for such surfaces over broader fields.

Findings

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

Sufficient conditions for unirationality over fields with characteristic not two.

Extension of previous partial results to all such surfaces.

Abstract

We prove that every del Pezzo surface of degree two over a finite field is unirational, building on the work of Manin and an extension by Salgado, Testa, and V\'arilly-Alvarado, who had proved this for all but three surfaces. Over general fields of characteristic not equal to two, we state sufficient conditions for a del Pezzo surface of degree two to be unirational.