TGiF-Seminar: Tropical geometry in Frankfurt (Second meeting Winter Semester 2022/23)

Navid Nabijou (University of Cambridge)

Abstract: I will discuss recent work concerning maps from tropical curves to orthants. A “combinatorial type” of such map is the data of an abstract graph together with slope vectors along the edges. To each such combinatorial type there is an associated moduli space, which parametrises metric enhancements of the graph compatible with the given slopes. This moduli space is a rational polyhedral cone, giving rise to an affine toric variety.
Our main result shows that every rational polyhedral cone appears as the moduli space associated to some combinatorial type of tropical map. This establishes universality (also known as Murphy’s law) for tropical maps. The proof is constructive and extremely concrete, as I will demonstrate. Combined with insights from logarithmic geometry, our result implies that every toric singularity appears as a virtual singularity on a moduli space of stable logarithmic maps.