Toric residue and combinatorial degree
Author:
Ivan Soprounov
Journal:
Trans. Amer. Math. Soc. 357 (2005), 19631975
MSC (2000):
Primary 14M25; Secondary 52B20
Published electronically:
October 7, 2004
MathSciNet review:
2115085
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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.
 [AK]
C. D'Andrea, A. Khetan, Macaulay style formulas for toric residues, preprint math.AG/0307154.
 [BM]
Victor
V. Batyrev and Evgeny
N. Materov, Toric residues and mirror symmetry, Mosc. Math. J.
2 (2002), no. 3, 435–475. Dedicated to Yuri I.
Manin on the occasion of his 65th birthday. MR 1988969
(2005a:14070)
 [CCD]
Eduardo
Cattani, David
Cox, and Alicia
Dickenstein, Residues in toric varieties, Compositio Math.
108 (1997), no. 1, 35–76. MR 1458757
(98f:14039), http://dx.doi.org/10.1023/A:1000180417349
 [CaD]
Eduardo
Cattani and Alicia
Dickenstein, A global view of residues in the torus, J. Pure
Appl. Algebra 117/118 (1997), 119–144. Algorithms
for algebra (Eindhoven, 1996). MR 1457836
(98i:14050), http://dx.doi.org/10.1016/S00224049(97)00008X
 [CD]
D. Cox, A. Dickenstein, A Codimension Theorem for Complete Toric Varieties, preprint, math.AG/0310108.
 [CDS]
Eduardo
Cattani, Alicia
Dickenstein, and Bernd
Sturmfels, Residues and resultants, J. Math. Sci. Univ. Tokyo
5 (1998), no. 1, 119–148. MR 1617074
(2000b:14065)
 [C1]
David
A. Cox, The homogeneous coordinate ring of a toric variety, J.
Algebraic Geom. 4 (1995), no. 1, 17–50. MR 1299003
(95i:14046)
 [C2]
David
A. Cox, Toric residues, Ark. Mat. 34 (1996),
no. 1, 73–96. MR 1396624
(97e:14062), http://dx.doi.org/10.1007/BF02559508
 [F]
William
Fulton, Introduction to toric varieties, Annals of Mathematics
Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993.
The William H. Roever Lectures in Geometry. MR 1234037
(94g:14028)
 [GKh1]
O.
A. Gel′fond and A.
G. Khovanskiĭ, Newton polyhedra and Grothendieck
residues, Dokl. Akad. Nauk 350 (1996), no. 3,
298–300 (Russian). MR 1444043
(98b:32004)
 [GKh2]
O.
A. Gelfond and A.
G. Khovanskii, Toric geometry and Grothendieck residues, Mosc.
Math. J. 2 (2002), no. 1, 99–112, 199 (English,
with English and Russian summaries). MR 1900586
(2003h:14074)
 [KS]
A. Khetan, I. Soprounov, Partition matrices for polytopes towards computing toric residue, preprint, math.AG/0406279.
 [Kh]
A.
Khovanskii, Newton polyhedra, a new formula for mixed volume,
product of roots of a system of equations, The Arnoldfest (Toronto,
ON, 1997) Fields Inst. Commun., vol. 24, Amer. Math. Soc.,
Providence, RI, 1999, pp. 325–364. MR 1733583
(2001d:52013)
 [M]
Anvar
R. Mavlyutov, The Hodge structure of semiample hypersurfaces and a
generalization of the monomialdivisor mirror map, Advances in
algebraic geometry motivated by physics (Lowell, MA, 2000), Contemp.
Math., vol. 276, Amer. Math. Soc., Providence, RI, 2001,
pp. 199–227. MR 1837119
(2002c:14067), http://dx.doi.org/10.1090/conm/276/04522
 [S]
I. Soprounov, Residues and tame symbols on toroidal varieties, to appear in Compositio Math., math.AG/0203114.
 [AK]
 C. D'Andrea, A. Khetan, Macaulay style formulas for toric residues, preprint math.AG/0307154.
 [BM]
 V. Batyrev, E. Materov, Toric Residues and Mirror Symmetry, Moscow Math. J. 2 (2002), no. 3, 435475. MR 1988969
 [CCD]
 E. Cattani, D. Cox, A. Dickenstein, Residues in Toric Varieties, Compositio Math. 108 (1997), no. 1, 3576. MR 1458757 (98f:14039)
 [CaD]
 E. Cattani, A. Dickenstein, A global view of residues in the torus, J. Pure Appl. Algebra 117/118 (1997), 119144. MR 1457836 (98i:14050)
 [CD]
 D. Cox, A. Dickenstein, A Codimension Theorem for Complete Toric Varieties, preprint, math.AG/0310108.
 [CDS]
 E. Cattani, A. Dickenstein, B. Sturmfels, Residues and Resultants, J. Math. Sci. Univ. Tokyo 5 (1998), 119148. MR 1617074 (2000b:14065)
 [C1]
 D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebr. Geom. 4 (1995), 1750. MR 1299003 (95i:14046)
 [C2]
 D. A. Cox, Toric residues, Arkiv für Matematik 34 (1996) 7396. MR 1396624 (97e:14062)
 [F]
 W. Fulton, Introduction to Toric Varieties, Princeton Univ. Press, Princeton, 1993. MR 1234037 (94g:14028)
 [GKh1]
 O. A. Gelfond and A. G. Khovanskii, Newton polyhedra and Grothendieck residues (in Russian), Dokl. Akad. Nauk, 350, no. 3 (1996), 298300. MR 1444043 (98b:32004)
 [GKh2]
 O. A. Gelfond, A. G. Khovanskii, Toric geometry and Grothendieck residues, Moscow Math. J. 2 (2002), no. 1, 99112. MR 1900586 (2003h:14074)
 [KS]
 A. Khetan, I. Soprounov, Partition matrices for polytopes towards computing toric residue, preprint, math.AG/0406279.
 [Kh]
 A. G. Khovanskii, Newton polyhedra, a new formula for mixed volume, product of roots of a system of equations, Fields Inst. Comm., Vol. 24 (1999), 325364. MR 1733583 (2001d:52013)
 [M]
 A. Mavlyutov, The Hodge structure of semiample hypersurfaces and a generalization of the monomialdivisor mirror map, Advances in algebraic geometry motivated by physics, Contemp. Math. 276 (2000). MR 1837119 (2002c:14067)
 [S]
 I. Soprounov, Residues and tame symbols on toroidal varieties, to appear in Compositio Math., math.AG/0203114.
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
14M25,
52B20
Retrieve articles in all journals
with MSC (2000):
14M25,
52B20
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/S0002994704037705
PII:
S 00029947(04)037705
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.
