Intersection theory of the stable pair compactification of the moduli space of six lines in the plane

TL;DR

This paper investigates the intersection theory of the stable pair compactification of the moduli space of six lines in the plane, providing explicit descriptions of Chow rings and resolutions, and extending classical constructions.

Contribution

It introduces blowup sequences generalizing Keel's and Kapranov's constructions, describes the Chow ring structure of the compactification, and extends intersection theory with higher-dimensional psi-classes.

Findings

Chow ring of small resolutions has Keel-like presentation

Chow ring of the compactification is an explicit subring of a resolution

Provides an independent proof that the space is the log canonical compactification

Abstract

We describe sequences of blowups of $\overline{M}_{0,5} \times \overline{M}_{0,5}$ and $\mathbf{P}^2 \times \mathbf{P}^2$ yielding a small resolution of the stable pair compactification $\overline{M}(3,6)$ of the moduli space $M(3,6)$ of six lines in $\mathbf{P}^2$. These blowup sequences can be viewed, respectively, as generalizations of Keel's and Kapranov's constructions of $\overline{M}_{0,n}$. We use these blowup sequences to describe the intersection theory of $\overline{M}(3,6)$. In particular, we show that the Chow ring of any small resolution of $\overline{M}(3,6)$ has a presentation analogous to Keel's presentation of $A^*(\overline{M}_{0,n})$, and the Chow ring of $\overline{M}(3,6)$ is an explicit subring of the Chow ring of one of these small resolutions. We also introduce higher-dimensional versions of the $\psi$-classes on $\overline{M}_{0,n}$, and describe their intersections on $\overline{M}(3,6)$. Finally, we use our results to obtain an independent proof of Luxton's result that $\overline{M}(3,6)$ is the log canonical compactification of $M(3,6)$.