Moduli of genus one curves with two marked points as a weighted blow-up

TL;DR

This paper provides an explicit geometric description of the moduli space of genus one curves with two marked points, and uses it to compute its Brauer group and Chow rings over suitable base schemes.

Contribution

It introduces a weighted blow-up model for ar{\u03bc}_{1,2} and applies this to compute its Brauer group and Chow rings, advancing understanding of its algebraic structure.

Findings

Explicit weighted blow-up description of ar{rac{1}{2}}

Computed Brauer group over schemes where 6 is invertible

Determined integral Chow rings of ar{rac{1}{2}} and rac{rac{1}{2}}

Abstract

We give an explicit description of $\overline{\mathcal{M}}_{1,2}$ as a weighted blow-up of a weighted projective stack. We use this description to compute the Brauer group of $\overline{\mathcal{M}}_{1,2;S}$ over any base scheme $S$ where 6 is invertible, and the integral Chow rings of $\overline{\mathcal{M}}_{1,2}$ and $\mathcal{M}_{1,2}$.