Rational curves on smooth cubic hypersurfaces over finite fields

TL;DR

This paper proves that for smooth cubic hypersurfaces over finite fields with characteristic >3 and dimension at least 11, the space of rational curves of fixed degree is irreducible and has the expected dimension.

Contribution

It establishes the irreducibility and expected dimension of rational curve spaces on high-dimensional smooth cubic hypersurfaces over finite fields.

Findings

Space of rational curves is irreducible.

Dimension matches the expected value.

Results hold for characteristic >3 and dimension ≥11.

Abstract

Let k be a finite field with characteristic exceeding 3. We prove that the space of rational curves of fixed degree on any smooth cubic hypersurface over k with dimension at least 11 is irreducible and of the expected dimension.