Quasilinear tropical compactifications

Abstract

The prototypical examples of tropical compactifications are compactifications of complements of hyperplane arrangements, which posses a number of remarkable properties not satisfied by more general tropical compactifications of closed subvarieties of tori. We introduce a broader class of tropical compactifications, which we call quasilinear (tropical) compactifications, and which continue to satisfy the desirable properties of compactifications of complements of hyperplane arrangements. In particular, we show any quasilinear compactification is sch\"on, and its intersection theory is described entirely by the intersection theory of the corresponding tropical fan. As applications, we prove the quasilinearity of the moduli spaces of 6 lines in $\mathbb{P}^2$ and marked cubic surfaces, obtaining results on the geometry of the stable pair compactifications of these spaces.