## Osaka Journal of Mathematics

### Facets of secondary polytopes and chow stability of toric varieties

Naoto Yotsutani

#### Abstract

Chow stability is one notion of Mumford's geometric invariant theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent secondary polytope, which is a polytope whose vertices correspond to regular triangulations of the associated polytope [7]. In this paper, we give a purely convex-geometrical proof that the Chow form of a projective toric variety is $H$-semistable if and only if it is $H$-polystable with respect to the standard complex torus action $H$. This essentially means that Chow semistability is equivalent to Chow polystability for any (not-necessaliry-smooth) projective toric varieties.

#### Article information

Source
Osaka J. Math., Volume 53, Number 3 (2016), 751-765.

Dates
First available in Project Euclid: 5 August 2016

https://projecteuclid.org/euclid.ojm/1470413988

Mathematical Reviews number (MathSciNet)
MR3533467

Zentralblatt MATH identifier
06629523

#### Citation

Yotsutani, Naoto. Facets of secondary polytopes and chow stability of toric varieties. Osaka J. Math. 53 (2016), no. 3, 751--765. https://projecteuclid.org/euclid.ojm/1470413988

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