On the Kodaira dimension of Hurwitz spaces

TL;DR

This paper proves that the Hurwitz stack for covers of the projective line with genus greater than 1 and degree greater than 2 has a big canonical bundle, and that all trigonal curve moduli spaces of genus greater than 1 are of general type.

Contribution

It establishes the bigness of the canonical bundle for Hurwitz stacks and the general type of trigonal curve moduli spaces, advancing understanding of their geometric properties.

Findings

Hurwitz stack's canonical bundle is big for genus g>1, degree k>2 covers.

All coarse moduli spaces of trigonal curves of genus g>1 are of general type.

Provides new insights into the Kodaira dimension of Hurwitz spaces.

Abstract

We show that the canonical bundle of the Hurwitz stack classifying covers of genus g>1 and degree k>2 of the projective line is big. We show that all coarse moduli spaces of trigonal curves of genus g>1 are of general type.