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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. document.getElementById('cloak25a5c03d75f158fa5b0ce938cef95827').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy25a5c03d75f158fa5b0ce938cef95827 = 's&#111;tt&#105;l&#101;' + '&#64;'; addy25a5c03d75f158fa5b0ce938cef95827 = addy25a5c03d75f158fa5b0ce938cef95827 + 'm&#97;th' + '&#46;' + 't&#97;m&#117;' + '&#46;' + '&#101;d&#117;'; var addy_text25a5c03d75f158fa5b0ce938cef95827 = 's&#111;tt&#105;l&#101;' + '&#64;' + 'm&#97;th' + '&#46;' + 't&#97;m&#117;' + '&#46;' + '&#101;d&#117;';document.getElementById('cloak25a5c03d75f158fa5b0ce938cef95827').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy25a5c03d75f158fa5b0ce938cef95827 + '\'>'+addy_text25a5c03d75f158fa5b0ce938cef95827+'<\/a>';

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).