Facets of secondary polytopes and Chow stability of toric varieties

TL;DR

This paper proves that for projective toric varieties, Chow semistability and polystability are equivalent, using convex geometry and secondary polytopes, which simplifies understanding their stability properties.

Contribution

It provides a convex-geometrical proof linking Chow stability to secondary polytopes, showing their equivalence for all projective toric varieties.

Findings

Chow semistability is equivalent to Chow polystability for projective toric varieties.

A convex-geometrical proof is established for the stability equivalence.

Secondary polytopes determine Chow stability of toric varieties.

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 {\it secondary polytope}, which is a polytope whose vertices correspond to regular triangulations of the associated polytope \cite{KSZ}. 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 \emph{essentially} means that Chow semistability is equivalent to Chow polystability for any (not-necessaliry-smooth) projective toric varieties.