Degree of Unirationality for del Pezzo Surfaces over Finite Fields

TL;DR

This paper investigates the degree of unirational parameterizations for degree three and four del Pezzo surfaces over finite fields, establishing new bounds and highlighting open questions in the field.

Contribution

It proves degree two parameterizations for degree four del Pezzo surfaces and degree six for minimal cubic surfaces over finite fields, advancing understanding of their unirationality.

Findings

Degree four del Pezzo surfaces admit degree two parameterizations.

Minimal cubic surfaces admit degree six parameterizations.

Open question on degree 3 or 4 parameterizations for minimal cubic surfaces.

Abstract

We address the question of the degree of unirational parameterizations of degree four and degree three del Pezzo surfaces. Specifically we show that degree four del Pezzo surfaces over finite fields admit degree two parameterizations and minimal cubic surfaces admit parameterizations of degree 6. It is an open question whether or not minimal cubic surfaces over finite fields can admit degree 3 or 4 parameterizations.