On the arithmetic of one del Pezzo surface over the field with three elements

TL;DR

This paper investigates the existence of rational curves on a specific del Pezzo surface over the finite field with three elements, using computational methods to identify all such curves of a certain degree.

Contribution

It introduces a computational algorithm to search for rational curves on the surface and provides explicit counts of these curves at degree 8.

Findings

920 rational curves of degree 8 found on the surface

No rational curves of smaller degree exist on the surface

Algorithm effectively enumerates rational curves on algebraic surfaces

Abstract

We discuss the problem of existence of rational curves on a certain del Pezzo surface from a computational point of view and suggest a computer algorithm implementing search. In particular, our computations reveal that the surface contains 920 rational curves with parametrizations of degree 8 and does not contain rational curves for a smaller degree.