Intersection theory of the stable pair compactification of the moduli space of six lines in the plane (Q2115332)

Let \(M(r,n)\) be the space of arrangements of \(n\) hyperplanes in \(\mathbb{P}^{r-1}\) in general position. When \(r = 2\), the space \(M(2,n)\), usually denoted \(M_{0,n}\), has a celebrated compactification due to Grothendieck and Knudsen: the moduli space \(\overline{M}_{0,n}\) of stable \(n\)-pointed rational curves. The generalization of this construction to higher dimensions has been studied in [\textit{P. Hacking} et al., J. Algebr. Geom. 15, No. 4, 657--680 (2006; Zbl 1117.14036)]; \(\overline{M}(r,n)\), the moduli space of stable hyperplane arrangements, being the higher-dimensional version of \(\overline{M}_{0,n}\) for \(r\geq 3\). However, very little is known about its geometry, and it is natural to ask to what extent a given result about \(\overline{M}_{0,n}\) generalizes to \(\overline{M}(r,n)\). The intersection theory of \(\overline{M}_{0,n}\) was studied by \textit{S. Keel} in [Trans. Am. Math. Soc. 330, No. 2, 545--574 (1992; Zbl 0768.14002)]. Motivated by the desire to understand the intersection theory of \(\overline{M}(r,n)\), this paper is devoted to generalizing Keel's results to the first nontrivial higher-dimensional case: the stable pair compactification \(\overline{M}(3, 6)\) of the moduli space \(M(3,6)\) of six lines in general position in the plane. This moduli space was first studied by Luxton using tropical geometry. The author's approach differs from Luxton's in that he only uses elementary algebraic geometry. The key to author's study of \(\overline{M}(3,6)\) is an explicit construction of one of the small resolutions \(\widetilde{M}_{1}(3,6)\) as a sequence of blowups of \(\mathbb{P}^{2}\times\mathbb{P}^{2}\) in Section 3 (Theorem 3.4). The core of the paper is contained in Sections 4, 5 and 6 where the author gives a complete description of the intersection theory of \(\overline{M}(3, 6)\). The main result (Theorem 4.1) states that the Chow rings of the small resolutions of \(\overline{M}(3, 6)\) all admit presentations entirely analogous to Keel's presentation of the Chow ring \(A^{*}(\overline{M}_{0,n})\). In order to prove it, the author shows that it is enough to prove this result holds for the small resolution \(\widetilde{M}_{1}(3,6)\). Afterwards, he proves the main result in two parts throughout Section 4, the presentation of \(A^{*}(\widetilde{M}_{1}(3,6))\) being the most involved part. As an application of his results, the author gives an independent proof of Luxton's main result in Section 7 which states that \(\overline{M}(3,6)\) is the log canonical compactification of \(M(3,6)\), relying only on explicit birational geometry.