Combinatorics of line arrangements on tropical cubic surfaces

Algebra Seminar
Monday, April 1, 2019 - 12:50pm for 1 hour (actually 50 minutes)
Skiles 005
Maria Angelica Cueto – Ohio State University – cueto.5@osu.edu
Yoav Len

The classical statement that there are 27 lines on every smooth cubic surface in $\mathbb{P}^3$ fails to hold under tropicalization: a tropical cubic surface in $\mathbb{TP}^3$ often contains infinitely many tropical lines. This pathology can be corrected by reembedding the cubic surface in $\mathbb{P}^{44}$ via the anticanonical bundle.

Under this tropicalization, the 27 classical lines become an arrangement of metric trees in the boundary of the tropical cubic surface in $\mathbb{TP}^{44}$. A remarkable fact is that this arrangement completely determines the combinatorial structure of the corresponding tropical cubic surface. In this talk, we will describe their metric and topological type as we move along the moduli space of tropical cubic surfaces. Time permitting, we will discuss the matroid that emerges from their tropical convex hull.

This is joint work with Anand Deopurkar.