Birational Contractions of $\bar{M}_{3,1}$ and $\bar{M}_{4,1}$

TL;DR

This paper investigates the birational geometry of moduli spaces of pointed curves of genus 3 and 4, proving a pointed Slope Conjecture and analyzing contractions that reveal extremal divisors in their effective cones.

Contribution

It introduces a pointed analogue of the Slope Conjecture and constructs explicit birational contractions using GIT, identifying extremal divisors in the effective cones of these moduli spaces.

Findings

Proves the pointed Slope Conjecture for genus 3 and 4.

Constructs birational contractions contracting pointed Brill-Noether divisors.

Shows these divisors generate extremal rays of the effective cones.

Abstract

We study the birational geometry of $\bar{M}_{3,1}$ and $\bar{M}_{4,1}$. In particular, we pose a pointed analogue of the Slope Conjecture and prove it in these low-genus cases. Using variation of GIT, we construct birational contractions of these spaces in which certain divisors of interest -- the pointed Brill-Noether divisors -- are contracted. As a consequence, we see that these pointed Brill-Noether divisors generate extremal rays of the effective cones for these spaces.