Sharp slope bounds for sweeping families of trigonal curves

TL;DR

This paper determines precise bounds on the slopes of sweeping families of trigonal curves in the moduli space, confirming a conjecture for even genus and providing explicit divisor expressions for both even and odd genus cases.

Contribution

It establishes sharp slope bounds for trigonal curve families in moduli space, confirming a conjecture for even genus and deriving explicit divisor formulas for both cases.

Findings

Proved the slope bound of 7+6/g for even g.

Derived the bound of 7+20/(3g+1) for odd g.

Explicit expressions for Maroni divisors in both cases.

Abstract

We establish sharp bounds for the slopes of curves in $\bar{M}_g$ that sweep the locus of trigonal curves, proving Stankova-Frenkel's conjectured bound of $7+6/g$ for even $g$ and obtaining the bound $7+20/(3g+1)$ for odd $g$. For even $g$, we find an explicit expression of the so-called Maroni divisor in the Picard group of the space of admissible triple covers. For odd $g$, we describe the analogous extremal effective divisor and give a similar explicit expression.