Combinatorial construction of toric residues.
Annales de l'Institut Fourier, Volume 55 (2005) no. 2, p. 511-548
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.
Dans cet article nous étudions l’existence d’un élément explicite dont le résidu torique est égal à un. On peut trouver un tel élément si et seulement si les polytopes associés sont essentiels. Nous réduisons ce problème à l’existence d’une collection de partitions des points du réseau dans les polytopes qui satisfont une certaine condition combinatoire. Nous utilisons cette description pour résoudre le problème pour n=2 et pour tout n si les polytopes des diviseurs ont en commun un drapeau complet de faces. Ceci généralise des résultats antérieurs dans le cas où les diviseurs sont amples.
DOI : https://doi.org/10.5802/aif.2106
Classification:  14M25,  52B20,  06A07
Keywords: Toric varieties, toric residues, semi-ample degrees, facet colorings, combinatorial degree
@article{AIF_2005__55_2_511_0,
     author = {Khetan, Amit and Soprounov, Ivan},
     title = {Combinatorial construction of toric residues.},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {55},
     number = {2},
     year = {2005},
     pages = {511-548},
     doi = {10.5802/aif.2106},
     mrnumber = {2147899},
     zbl = {1077.14073},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2005__55_2_511_0}
}
Khetan, Amit; Soprounov, Ivan. Combinatorial construction of toric residues.. Annales de l'Institut Fourier, Volume 55 (2005) no. 2, pp. 511-548. doi : 10.5802/aif.2106. https://aif.centre-mersenne.org/item/AIF_2005__55_2_511_0/

[1] Ian Anderson A first course in combinatorial mathematics (2nd ed.), Oxford University Press (1989) | MR 1029023 | Zbl 0662.05002

[2] V. Batyrev; D. Cox On the Hodge structure of projective hypersurfaces in toric varieties, Duke J. Math., Tome 75 (1994), pp. 293-338 | MR 1290195 | Zbl 0851.14021

[3] V. Batyrev; E. Materov Toric Residues and Mirror Symmetry, Moscow Math. J., Tome 2 (2002) no. 3, pp. 435-475 | MR 1988969 | Zbl 1026.14016

[4] E. Cattani; D. Cox; A. Dickenstein Residues in Toric Varieties, Compositio Math., Tome 108 (1997) no. 1, pp. 35-76 | Article | MR 1458757 | Zbl 0883.14029

[5] E. Cattani; A. Dickenstein A global view of residues in the torus, J. Pure Appl. Algebra, Tome 117/118 (1997), pp. 119-144 | Article | MR 1457836 | Zbl 0899.14024

[6] E. Cattani; A. Dickenstein Planar Configurations of Lattice Vectors and GKZ-Rational Toric Fourfolds in 6 , J. Alg. Comb., Tome 19 (2004), pp. 47-65 | Article | MR 2056766 | Zbl 1054.33009

[7] E. Cattani; A. Dickenstein; B. Sturmfels Residues and Resultants, J. Math. Sci. Univ. Tokyo, Tome 5 (1998), pp. 119-148 | MR 1617074 | Zbl 0933.14033

[8] E. Cattani; A. Dickenstein; B. Sturmfels Computing multidimensional residues, Birkhäuser, Basel (Progress in Math.) (1996), pp. 135-164 | Zbl 0882.13020

[9] E. Cattani; A. Dickenstein; B. Sturmfels Rational hypergeometric functions, Compositio Math., Tome 128 (2001), pp. 217-240 | Article | MR 1850183 | Zbl 0990.33013

[10] E. Cattani; A. Dickenstein; B. Sturmfels Binomial Residues, Ann. Inst. Fourier, Tome 52 (2002), pp. 687-708 | Article | Numdam | MR 1907384 | Zbl 1015.32007

[11] D.A. Cox; A. Dickenstein Codimension theorems for complete toric varieties (to appear in Proc. AMS, math.AG/0310108) | MR 2160176 | Zbl 1083.14058 | Zbl 02188235

[12] D.A. Cox The homogeneous coordinate ring of a toric variety, J. Alg. Geom., Tome 4 (1995), pp. 17-50 | MR 1299003 | Zbl 0846.14032

[13] D.A. Cox Toric residues, Arkiv Mat., Tome 34 (1996), pp. 73-96 | Article | MR 1396624 | Zbl 0904.14029

[14] C. D'Andrea; A. Khetan Macaulay style formulas for toric residues (to appear in Compositio Math., math.AG/0307154) | MR 2135285 | Zbl 1076.14081 | Zbl 02183037

[15] W. Fulton Introduction to Toric Varieties, Princeton Univ. Press, Princeton (1993) | MR 1234037 | Zbl 0813.14039

[16] I.M. Gelfand; M.M. Kapranov; A.V. Zelevinsky Discriminants, resultants, and multidimensional determinants, Birkhäuser Boston, Inc., Boston (1994) | MR 1264417 | Zbl 0827.14036

[17] O.A. Gelfond; A.G. Khovanskii Toric geometry and Grothendieck residues, Moscow Math. J., Tome 2 (2002) no. 1, pp. 99-112 | MR 1900586 | Zbl 1044.14029

[18] I. Soprounov Residues and tame symbols on toroidal varieties, Compositio Math., Tome 140 (2004) no. 6, pp. 1593-1613 | MR 2098404 | Zbl 1075.14052 | Zbl 02140866

[19] I. Soprounov Toric residue and combinatorial degree, Trans. Amer. Math. Soc., Tome 357 (2005) no. 5, pp. 1963-1975 | Article | MR 2115085 | Zbl 1070.14048 | Zbl 02140161