Contracting the Weierstrass locus to a point

TL;DR

The paper constructs a specific open substack of the moduli space of genus g pointed curves that contracts the Weierstrass locus to a point, using alternative birational models.

Contribution

It introduces a new open substack and a morphism that simplifies the Weierstrass locus within the moduli space of curves.

Findings

Weierstrass locus is contracted to a point in the constructed substack.

The morphism is an isomorphism outside the Weierstrass locus.

Uses alternative birational models of moduli spaces.

Abstract

We construct an open substack $U\subset\mathcal{M}_{g,1}$ with the complement of codimension $\ge 2$ and a morphism from $U$ to a weighted projective stack, which sends the Weierstrass locus $\mathcal{W}\cap U$ to a point, and maps $\mathcal{M}_{g,1}\setminus\mathcal{W}$ isomorphically to its image. The proof uses alternative birational models of $\mathcal{M}_{g,1}$ and $\mathcal{M}_{g,2}$ from arXiv:1509.07241.