Title: Toric residue and combinatorial degree [PDF]

Author: Ivan Soprounov

Abstract: 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 , . . . , 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.