Moduli of Trigonal Curves

TL;DR

This paper investigates the moduli space of trigonal curves, establishing bounds on the slope of trigonal fibrations, analyzing associated vector bundles, and describing the Picard group of the trigonal locus.

Contribution

It provides the exact upper bound for the slope of trigonal fibrations and relates Bogomolov-semistability to geometric divisors in the moduli space.

Findings

Established the upper bound ${36(g+1)}/(5g+1)$ for the slope.

Linked Bogomolov-semistability to the Maroni divisor in even genus cases.

Described the rational Picard group of the trigonal locus in the moduli space.

Abstract

We study the moduli of trigonal curves. We establish the exact upper bound of ${36(g+1)}/(5g+1)$ for the slope of trigonal fibrations. Here, the slope of any fibration $X\to B$ of stable curves with smooth general member is the ratio $\delta_B/\lambda_B$ of the restrictions of the boundary class $\delta$ and the Hodge class $\lambda$ on the moduli space $\bar{\mathfrak{M}}_g$ to the base $B$. We associate to a trigonal family $X$ a canonical rank two vector bundle $V$, and show that for Bogomolov-semistable $V$ the slope satisfies the stronger inequality ${\delta_B}/{\lambda_B}\leq 7+{6}/{g}$. We further describe the rational Picard group of the {trigonal} locus $\bar{\mathfrak T}_g$ in the moduli space $\bar{\mathfrak{M}}_g$ of genus $g$ curves. In the even genus case, we interpret the above Bogomolov semistability condition in terms of the so-called Maroni divisor in $\bar{\mathfrak T}_g$.