Birational models of ${\mathcal M}_{2,2}$ arising as moduli of curves with nonspecial divisors

TL;DR

This paper explores birational models of the moduli space of genus 2 curves with 2 marked points, focusing on nonspecial divisors, and establishes their geometric properties and projectivity.

Contribution

It describes specific birational models of ${ m M}_{2,2}$ derived from nonspecial divisors and proves the projectivity of the associated moduli space.

Findings

Identification of singular curves in the models

Construction of a model via blowing down the Weierstrass divisor

Proof of projectivity for the coarse moduli space

Abstract

We study birational projective models of ${\mathcal M}_{2,2}$ obtained from the moduli space of curves with nonspecial divisors. We describe geometrically which singular curves appear in these models and show that one of them is obtained by blowing down the Weierstrass divisor in the moduli stack of ${\mathcal Z}$-stable curves $\overline{\mathcal M}_{2,2}({\mathcal Z})$ defined by Smyth. As a corollary, we prove projectivity of the coarse moduli space $\overline{M}_{2,2}({\mathcal Z})$.