Some more geometry in the secondary fan

Frank Sottile, Texas A&M University, College Station

January 6, 2015

A polyhedral subdivision of finitely many points $A$ in $\mathbb{R}^d$ is a decomposition of its convex hull into non-overlapping polytopes with vertices in $A$. It is regular if the polytopes are projections of facets of a polyhedron in $\mathbb{R}^{d+1}$. Gelfand, Kapranov, and Zelevinsky introduced the secondary fan of $A$, which encodes all regular subdivisions of $A$. I will talk on work with Postinghel and Villamizar that explains another geometric interpretation of the secondary fan. The points $A$ give a curvilinear copy of the convex hull of A in the probability simplex whose vertices correspond to $A$ which we call the real toric variety of $A$. We consider possible Hausdorff limits of translations of this real toric variety. We show that the possible limits correspond to the cones of the secondary fan. Our main tool is a study of sequences in the ambient space of the secondary fan.