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 $A^{\ast }({\overline{M}}_{0,n})$, 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 $M(3,6)$.

34 pages, 1 figure