Modular compactifications of the space of marked trigonal curves

TL;DR

This paper develops a sequence of modular compactifications for the space of marked trigonal curves, involving contractions and flips, leading to Fano models and fibrations, and explicitly describes the associated Mori chamber decomposition.

Contribution

It introduces a novel sequence of compactifications using contractions and flips tailored to trigonal curves, including explicit Mori chamber descriptions.

Findings

Constructed a sequence of compactifications with boundary modifications.

Identified Fano models and fibrations for different genera.

Explicit Mori chamber decomposition for the sequence of flips.

Abstract

We construct a sequence of modular compactifications of the space of marked trigonal curves by allowing the branch points to coincide to a given extent. Beginning with the standard admissible cover compactification, the sequence first proceeds through contractions of the boundary divisors and then through flips of the so-called Maroni strata, culminating in a Fano model for even genera and a Fano fibration for odd genera. While the sequence of divisorial contractions arises from a more general construction, the sequence of flips uses the particular geometry of triple covers. We explicitly describe the Mori chamber decomposition given by this sequence of flips.