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Facets of secondary polytopes and Chow stability of toric varieties. (English) Zbl 1454.14125
Summary: Chow stability is one notion of Mumford’s geometric invariant theory for studying the moduli space of polarized varieties. M. M. Kapranov et al. 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 [Duke Math. J. 67, No. 1, 189–218 (1992; Zbl 0780.14027)]. 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.

##### MSC:
 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14L24 Geometric invariant theory 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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