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

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 $M(3,6)$.