Rational surfaces on low degree hypersurfaces

TL;DR

This paper applies function field analytic number theory to study the moduli space of morphisms from the projective plane to smooth hypersurfaces, focusing on irreducibility and dimension for low-degree cases.

Contribution

It establishes irreducibility and dimension of the moduli space of morphisms to smooth hypersurfaces of small degree using novel number theoretic techniques.

Findings

Moduli space is irreducible for low-degree hypersurfaces.

Dimension of the moduli space is explicitly determined.

Method combines algebraic geometry with analytic number theory.

Abstract

We use function field analytic number theory to establish the irreducibility and dimension of the moduli space that parameterises morphisms of fixed degree from $\mathbb{P}^2$ to an arbitrary smooth hypersurface of sufficiently small degree.