Gale duality and homogeneous toric varieties

ABSTRACT

A non-degenerate toric variety X is called S-homogeneous if the subgroup of the automorphism group Aut(X) generated by root subgroups acts on X transitively. We prove that maximal S-homogeneous toric varieties are in bijection with pairs (P,𝒜), where P is an abelian group and 𝒜 is a finite collection of elements in P such that 𝒜 generates the group P and for every a∈𝒜 the element a is contained in the semigroup generated by 𝒜∖{a}. We show that any non-degenerate homogeneous toric variety is a big open toric subset of a maximal S-homogeneous toric variety. In particular, every homogeneous toric variety is quasiprojective. We conjecture that any non-degenerate homogeneous toric variety is S-homogeneous.

KEYWORDS:

• Automorphism
• Demazure root
• divisor class group
• Gale duality
• toric variety

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 14J50
• 14M17
• Secondary: 13A50
• 14L30
• 14R20