Geometry of tropical compactifications of moduli spaces

We develop techniques for studying tropical compactifications of closed subvarieties of tori by introducing a broad class of such tropical compactifications, called \emph{quasilinear} tropical compactifications, which satisfy a number of remarkable properties generalizing compactifications of complements of hyperplane arrangements. We apply these techniques to study the birational geometry and intersection theory of certain compactifications of moduli spaces, namely, the moduli spaces $M(r,n)$ of hyperplane arrangements and $Y(3,n)$ of marked del Pezzo surfaces. In particular, we prove a conjecture of Keel and Tevelev that the stable pair compactification of $M(r,n)$ for $r=2$ or $r=3$ and $n \leq 8$ is the log canonical compactification, and we describe the intersection theory and cohomology of tropical compactifications of $M(r,n)$ for $r=2$, or $r=3$, $n \leq 8$ and tropical compactifications of $Y(3,n)$ for $n \leq 7$.