Rational curves on cubic hypersurfaces in positive characteristic

TL;DR

This paper investigates the moduli spaces of rational curves on cubic hypersurfaces over fields with characteristic not equal to 2 or 3, establishing irreducibility results for certain dimensions.

Contribution

It proves the irreducibility of the Kontsevich moduli space of stable maps on smooth cubic hypersurfaces of degree d in dimensions at least 4.

Findings

Kontsevich moduli space is irreducible for hypersurfaces of dimension ≥ 4

Results hold in characteristic not equal to 2 or 3

Applicable to all degrees d ≥ 1

Abstract

We study the moduli spaces of rational curves on cubic hypersurfaces in characteristic $\neq2,3$. As a result, we prove that for every integer $d\geq1$ the Kontsevich moduli space of stable maps on a smooth cubic hypersurface $X$ of degree $d$ is irreducible if the dimension of $X$ is greater than or equal to $4$.