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Session S31 - Mathematical-Physical Aspects of Toric and Tropical Geometry

Tuesday, July 13, 13:30 ~ 14:30 UTC-3

Irrational toric varieties and the secondary polytope

Frank Sottile

Texas A&M University, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Classical toric varieties come in two flavours: Normal toric varieties are given by rational fans in ${\mathbb R}^n$. A (not necessarily normal) affine toric variety is given by finite subset $A$ of ${\mathbb Z}^n$. When $A$ is homogeneous, it is projective. Applications of mathematics have long studied the positive real part of a toric variety as the main object, where the points $A$ may be arbitrary points in ${\mathbb R}^n$. For example, in 1963 Birch showed that such an irrational toric variety is homeomorphic to the convex hull of the set $A$.

Recent work showing that all Hausdorff limits of translates of irrational toric varieties are toric degenerations suggested the need for a theory of irrational toric varieties associated to arbitrary fans in ${\mathbb R}^n$. These are ${\mathbb R}^n_>$-equivariant cell complexes dual to the fan. Among the pleasing parallels with the classical theory is that the space of Hausdorff limits of the irrational projective toric variety of a finite set $A$ in ${\mathbb R}^n$ is homeomorphic to the secondary polytope of $A$.

This talk will sketch this story of irrational toric varieties. It represents work with Garcia-Puente, Zhu, Postinghel, Villamizar, and Pir.

Joint work with Luis David Garcia-Puente (Sam Houston State, USA), Ata Pir (TAMU, USA), Elisa Postinghel (Loughborough, UK), Nelly Villamizar (Swansea, UK) and Chungang Zhu (Dalian Univ. of Tech., China).

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