Wall-crossing integral Chow rings of $\overline{\mathcal M}_{1,n}$

TL;DR

This paper computes the integral Chow rings of the moduli stacks ,n for n=3,4, using blow-up sequences and inductive methods on alternative compactifications, advancing understanding of their intersection theory.

Contribution

It introduces a method to compute integral Chow rings of ,n by relating them to simpler stacks via blow-ups and blow-downs, extending previous work on compactifications.

Findings

Computed integral Chow rings for ,3 and ,4.

Established a recursive approach using blow-up sequences.

Connected the Chow rings to alternative compactifications.

Abstract

We compute the integral Chow rings of $\overline{\mathcal M}_{1,n}$ for $n=3,4$. For $n\leq 6$, these stacks can be obtained by a sequence of weighted blow-ups and blow-downs from a simple stack, either a weighted projective space or a Grassmannian. Our strategy consists in inductively computing all the integral Chow rings of the alternative compactifications introduced by Smyth and studied by Lekili-Polishchuk.