Degenerations in tropical compactifications and tropical intersection theory of ${\overline{M}}_{0,n}$

Abstract

The main result of this paper is a formula for the limit cycle of a 1-parameter family of subvarieties of a tropical compactification, expressed in terms of tropical intersections. Our theorem generalizes results of Dickenstein-Feichtner-Sturmfels and Katz to the case of tropical compactifications. In the second part of the paper, we apply our formula to the moduli space ${\overline{M}}_{0,n}$ of stable marked rational curves. We describe the tropicalization of the Kapranov maps ${\overline{M}}_{0,n}\to {\mathbb{P}}^{n-3}$, whose hyperplane pullbacks are the $\ddot{I}ˆ$-classes, with respect to a suitable choice of torus. We introduce tropical $\ddot{I}ˆ$-hypersurfaces (in genus zero). These are different from the standard definition of Mikhalkin and Kerber-Markwig, and may be of independent interest. We demonstrate our main result by giving a "firework algorithm" that computes limits of intersections of $\ddot{I}ˆ$-hypersurfaces.

50 pages, 10 figures, comments welcome