Rational curves on cubic hypersurfaces over finite fields

TL;DR

This paper develops a function field analogue of the Hardy-Littlewood circle method to count rational curves on smooth cubic hypersurfaces over finite fields, establishing their moduli space properties.

Contribution

It introduces a novel application of the circle method in the finite field setting to analyze rational curves on cubic hypersurfaces, including asymptotic counts and moduli space properties.

Findings

Asymptotic formula for the number of rational curves passing through two points

Dimension and irreducibility results for the moduli space of such curves

Applicability for large degree d

Abstract

Given a smooth cubic hypersurface $X$ over a finite field of characteristic greater than 3 and two generic points on $X$, we use a function field analogue of the Hardy-Littlewood circle method to obtain an asymptotic formula for the number of degree $d$ rational curves on $X$ passing through those two points. We use this to deduce the dimension and irreducibility of the moduli space parametrising such curves, for large enough $d$.