# Fields of definition of curves of a given degree

## TL;DR

This paper explores the Galois-module structure of rational curves of a given degree through specific points, relating it to deck transformations, and extends the analysis to hypersurfaces of low degree.

## Contribution

It introduces initial investigations into the Galois-module structure of rational curves and generalizes results to low-degree hypersurfaces.

## Key findings

- Preliminary insights into Galois-module structures.
- Relation between Galois modules and deck transformations.
- Asymptotic analysis of rational points on hypersurfaces.

## Abstract

Kontsevich and Manin gave a formula for the number $N_e$ of rational plane curves of degree $e$ through $3e-1$ points in general position in the plane. When these $3e-1$ points have coordinates in the rational numbers, the corresponding set of $N_e$ rational curves has a natural Galois-module structure. We make some extremely preliminary investigations into this Galois module structure, and relate this to the deck transformations of the generic fibre of the product of the evaluation maps on the moduli space of maps. We then study the asymptotics of the number of rational points on hypersurfaces of low degree, and use this to generalise our results by replacing the projective plane by such a hypersurface.