Toric residue and combinatorial degree

Author:
Ivan Soprounov

Journal:
Trans. Amer. Math. Soc. **357** (2005), 1963-1975

MSC (2000):
Primary 14M25; Secondary 52B20

DOI:
https://doi.org/10.1090/S0002-9947-04-03770-5

Published electronically:
October 7, 2004

MathSciNet review:
2115085

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Abstract: Consider an -dimensional projective toric variety defined by a convex lattice polytope . David Cox introduced the toric residue map given by a collection of divisors on . In the case when the are -invariant divisors whose sum is , 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 to the boundary of a simplex. This degree can be computed combinatorially. We also study radical monomial ideals of the homogeneous coordinate ring of . We give a necessary and sufficient condition for a homogeneous polynomial of semiample degree to belong to 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.

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Additional Information

**Ivan Soprounov**

Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003

Email:
isoprou@math.umass.edu

DOI:
https://doi.org/10.1090/S0002-9947-04-03770-5

Keywords:
Toric residues,
combinatorial degree,
toric variety,
homogeneous coordinate ring,
semiample degree

Received by editor(s):
October 19, 2003

Published electronically:
October 7, 2004

Article copyright:
© Copyright 2004
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.