2003/09/30 by Ivan Soprounov
Mathematics · #math.AG #msc:14M25 #msc:52B20
published as Trans. Amer. Math. Soc., 357 (2005), no. 5, 1963--1975 · 13 pages, one section added, 1 pstex figure. To appear in Trans. Amer. Math. Soc
arxiv created 2004/06/16 · arxiv updated 2009/12/01
Consider an n-dimensional projective toric variety X defined by a convex lattice polytope P. David Cox introduced the toric residue map given by a collection of n+1 divisors Z0,...,Zn on X. In the case when the Zi are T-invariant divisors whose sum is X\T the toric residue map is the multiplication by an integer number. We show that this number is the degree of a certain map from the boundary of the polytope P to the boundary of a simplex. This degree can be computed combinatorially. We also study radical monomial ideals I of the homogeneous coordinate ring of X. We give a necessary and sufficient condition for a homogeneous polynomial of semiample degree to belong to I in terms of geometry of toric varieties and combinatorics of fans. Both results have applications to the problem of constructing an element of residue one for semiample degrees.