Stable cohomology of the moduli space of trigonal curves

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.

24 pages, comments are welcome. v2: Proof of Lemma 4.2 corrected. v3: typos corrected

References in corpus (1)

• Rational cohomology of the moduli space of trigonal curves of genus 5