Combinatorial construction of toric residues.

Amit Khetan[1]; Ivan Soprounov

  • [1] University of Massachusetts, Department of Mathematics and Statistics, Amherst, MA 01003 (USA)

Annales de l’institut Fourier (2005)

  • Volume: 55, Issue: 2, page 511-548
  • ISSN: 0373-0956

Abstract

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In this paper we investigate the problem of finding an explicit element whose toric residue is equal to one. Such an element is shown to exist if and only if the associated polytopes are essential. We reduce the problem to finding a collection of partitions of the lattice points in the polytopes satisfying a certain combinatorial property. We use this description to solve the problem when n = 2 and for any n when the polytopes of the divisors share a complete flag of faces. The latter generalizes earlier results when the divisors were all ample.

How to cite

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Khetan, Amit, and Soprounov, Ivan. "Combinatorial construction of toric residues.." Annales de l’institut Fourier 55.2 (2005): 511-548. <http://eudml.org/doc/116199>.

@article{Khetan2005,
abstract = {In this paper we investigate the problem of finding an explicit element whose toric residue is equal to one. Such an element is shown to exist if and only if the associated polytopes are essential. We reduce the problem to finding a collection of partitions of the lattice points in the polytopes satisfying a certain combinatorial property. We use this description to solve the problem when $n=2$ and for any $n$ when the polytopes of the divisors share a complete flag of faces. The latter generalizes earlier results when the divisors were all ample.},
affiliation = {University of Massachusetts, Department of Mathematics and Statistics, Amherst, MA 01003 (USA)},
author = {Khetan, Amit, Soprounov, Ivan},
journal = {Annales de l’institut Fourier},
keywords = {Toric varieties; toric residues; semi-ample degrees; facet colorings; combinatorial degree; toric variety; semi-ample divisor},
language = {eng},
number = {2},
pages = {511-548},
publisher = {Association des Annales de l'Institut Fourier},
title = {Combinatorial construction of toric residues.},
url = {http://eudml.org/doc/116199},
volume = {55},
year = {2005},
}

TY - JOUR
AU - Khetan, Amit
AU - Soprounov, Ivan
TI - Combinatorial construction of toric residues.
JO - Annales de l’institut Fourier
PY - 2005
PB - Association des Annales de l'Institut Fourier
VL - 55
IS - 2
SP - 511
EP - 548
AB - In this paper we investigate the problem of finding an explicit element whose toric residue is equal to one. Such an element is shown to exist if and only if the associated polytopes are essential. We reduce the problem to finding a collection of partitions of the lattice points in the polytopes satisfying a certain combinatorial property. We use this description to solve the problem when $n=2$ and for any $n$ when the polytopes of the divisors share a complete flag of faces. The latter generalizes earlier results when the divisors were all ample.
LA - eng
KW - Toric varieties; toric residues; semi-ample degrees; facet colorings; combinatorial degree; toric variety; semi-ample divisor
UR - http://eudml.org/doc/116199
ER -

References

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  15. W. Fulton, Introduction to Toric Varieties, (1993), Princeton Univ. Press, Princeton Zbl0813.14039MR1234037
  16. I.M. Gelfand, M.M. Kapranov, A.V. Zelevinsky, Discriminants, resultants, and multidimensional determinants, (1994), Birkhäuser Boston, Inc., Boston Zbl0827.14036MR1264417
  17. O.A. Gelfond, A.G. Khovanskii, Toric geometry and Grothendieck residues, Moscow Math. J. 2 (2002), 99-112 Zbl1044.14029MR1900586
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  19. I. Soprounov, Toric residue and combinatorial degree, Trans. Amer. Math. Soc. 357 (2005), 1963-1975 Zbl1070.14048MR2115085

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