On the rationality of the moduli space of L\"uroth quartics

TL;DR

This paper proves that the moduli space of L"uroth quartics and the related space of Bateman seven-tuples in P^2 are both rational, revealing their algebraic structure and parametrization properties.

Contribution

It establishes the rationality of the moduli space of L"uroth quartics and Bateman seven-tuples, advancing understanding of their geometric and algebraic properties.

Findings

The moduli space M_L of L"uroth quartics is rational.

The moduli space of Bateman seven-tuples is rational.

Both spaces are algebraically parametrizable.

Abstract

We prove that the moduli space M_L of L"uroth quartics in P^2, i.e. the space of quartics which can be circumscribed around a complete pentagon of lines modulo the action of PGL_3(CC) is rational, as is the related moduli space of Bateman seven-tuples of points in P^2.