Toric residue and combinatorial degree
Abstract
Description
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 Z_0,...,Z_n on X. In the case when the Z_i 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.
13 pages, one section added, 1 pstex figure. To appear in Trans. Amer. Math. Soc
13 pages, one section added, 1 pstex figure. To appear in Trans. Amer. Math. Soc