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.Comment: 13 pages, one section added, 1 pstex figure. To appear in Trans.
Amer. Math. So
