Quasilinear tropical compactifications

TL;DR

This paper introduces quasilinear tropical compactifications, a new class that generalizes hyperplane arrangement complements, maintaining key properties and enabling detailed intersection theory and applications to moduli spaces.

Contribution

The paper defines quasilinear tropical compactifications, proves they are sch"on, and links their intersection theory to tropical fans, extending the class of well-understood compactifications.

Findings

Quasilinear compactifications are sch"on.

Their intersection theory is governed by tropical fans.

Applied to moduli spaces of lines and cubic surfaces.

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.