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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Toric residue and combinatorial degree


Author: Ivan Soprounov
Journal: Trans. Amer. Math. Soc. 357 (2005), 1963-1975
MSC (2000): Primary 14M25; Secondary 52B20
Published electronically: October 7, 2004
MathSciNet review: 2115085
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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 $(Z_0,\dots,Z_n)$ on $X$. In the case when the $Z_i$ are $\mathbb{T}$-invariant divisors whose sum is $X\setminus\mathbb{T}$, 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.


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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: http://dx.doi.org/10.1090/S0002-9947-04-03770-5
PII: S 0002-9947(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.