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Some more geometry in the secondary fan

Algebra & Discrete Mathematics

Speaker: Frank Sottile, Texas A&M and Simons Institute
Location: 1147 MSB
Start time: Mon, Nov 17 2014, 1:10PM

A polyhedral subdivision of finitely many points A in 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 R^{d+1}. Gelfand, Kapranov, and Zelevinsky introduced the secondary fan of A, which encodes all regular subdivisions of A. When A is integral, Kapranov, Sturmfels, and Zelevinsky showed how this fan encodes all possible limiting positions of a projective toric variety associated to A, under translation by the ambient torus.

I will talk on work with Postinghel and Villamizar that generalizes this to the situation where the points of A are not necessarily integral. Here, the toric variety is replaced by a curvilinear copy of the convex hull in the simplex. In this case, one does not have the tools from algebraic geometry (Hilbert schemes), and we instead develop tools based on the geometry of sequences of points in the secondary fan. This work was motivated by questions in geometric modeling and algebraic statistics which I plan to mention.
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