Toric difference variety

Toric difference variety

In this paper, the concept of toric difference varieties is defined and four equivalent descriptions for toric difference varieties are presented in terms of difference rational parametrization, difference coordinate rings, toric difference ideals, and group actions by difference tori. Connections between toric difference varieties and affine N[x]-semimodules are established by proving the corr...

The Signature of a Toric Variety

We identify a combinatorial quantity (the alternating sum of the h-vector) defined for any simple polytope as the signature of a toric variety. This quantity was introduced by R. Charney and M. Davis in their work, which in particular showed that its nonnegativity is closely related to a conjecture of H. Hopf on the Euler characteristic of a nonpositively curved manifold. We prove positive (or ...

The Functor of a Smooth Toric Variety

(1) Y 7→ {line bundle quotients of O Y } . This is easy to prove since a surjection O Y → L gives n + 1 sections of L which don’t vanish simultaneously and hence determine a map Y → Pnk . The goal of this paper is to generalize this representation to the case of an arbitrary smooth toric variety. We will work with schemes over an algebraically closed field k of characteristic zero, and we will ...

The Spectral Density Function of a Toric Variety

For a Kähler manifold X,ω with a holomorphic line bundle L and metric h such that the Chern form of L is ω, the spectral measures are the measures μN = P |sN,i| ν, where {sN,i}i is an L -orthonormal basis for H(X,L ), and ν is Liouville measure. We study the asymptotics in N of μN for X,L a Hamiltonian toric manifold, and give a very precise expansion in terms of powers 1/N and data on the mome...

Producing Good Quotients by Embedding into a Toric Variety