Stable cohomology of the moduli space of trigonal curves

TL;DR

This paper proves the stability of rational cohomology for the moduli space of trigonal curves in certain degrees, and computes the stable cohomology ring, showing it is isomorphic to the tautological ring.

Contribution

It establishes the stability range for the rational cohomology of trigonal curve moduli spaces and explicitly computes the stable cohomology ring.

Findings

Rational cohomology is independent of genus in degrees less than g/4.

Stable cohomology ring is isomorphic to the tautological ring.

Method involves embedding trigonal curves in Hirzebruch surfaces and using Gorinov-Vassiliev's technique.

Abstract

We prove that the rational cohomology $H^i(\mathcal{T}_g;\mathbf{Q})$ of the moduli space of trigonal curves of genus $g$ is independent of $g$ in degree $i<\lfloor g/4\rfloor.$ This makes possible to define the stable cohomology ring as $H^\bullet(\mathcal{T}_g;\mathbf{Q})$ for a sufficiently large $g.$ We also compute the stable cohomology ring, which turns out to be isomorphic to the tautological ring. This is done by studying the embedding of trigonal curves in Hirzebruch surfaces and using Gorinov-Vassiliev's method.