Birational properties of some moduli spaces related to tetragonal curves of genus 7

TL;DR

This paper proves that the moduli space of genus 7 curves with a g^1_4 and up to 11 marked points is rational, revealing new birational properties of these moduli spaces.

Contribution

It establishes the rationality of the locus of genus 7 curves with a g^1_4 for n up to 11, a new result in the birational geometry of these moduli spaces.

Findings

M^1_{7,n;4} is irreducible of dimension 17+n

M^1_{7,n;4} is rational for 0<= n <= 11

Provides new insights into the birational properties of moduli spaces of tetragonal curves

Abstract

Let M_{7,n} be the (coarse) moduli space of smooth curves of genus 7 with n marked points defined over the complex field. We denote by M^1_{7,n;4} the locus of points inside M_{7,n} representing curves carrying a g^1_4. It is classically known that M^1_{7,n;4} is irreducible of dimension 17+n. We prove in this paper that M^1_{7,n;4} is rational for 0<= n <= 11.