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The rank of a hypergeometric system

Part of: Semigroups Special varieties Homological methods Differential algebra

Published online by Cambridge University Press:  17 August 2010

Christine Berkesch
Christine Berkesch*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA (email: [email protected])
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Abstract

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The holonomic rank of the A-hypergeometric system MA(β) is the degree of the toric ideal IA for generic parameters; in general, this is only a lower bound. To the semigroup ring of A we attach the ranking arrangement and use this algebraic invariant and the exceptional arrangement of non-generic parameters to construct a combinatorial formula for the rank jump of MA(β). As consequences, we obtain a refinement of the stratification of the exceptional arrangement by the rank of MA(β) and show that the Zariski closure of each of its strata is a union of translates of linear subspaces of the parameter space. These results hold for generalized A-hypergeometric systems as well, where the semigroup ring of A is replaced by a non-trivial weakly toric module M⊆ℂ[ℤA] . We also provide a direct proof of the main result in [M. Saito, Isomorphism classes of A-hypergeometric systems, Compositio Math. 128 (2001), 323–338] regarding the isomorphism classes of MA (β) .

Keywords

hypergeometricD-moduletoricholonomicrankEuler–Koszulcombinatorial

MSC classification

Secondary: 33C70: Other hypergeometric functions and integrals in several variables 14M25: Toric varieties, Newton polyhedra 16E30: Homological functors on modules (Tor, Ext, etc.) 20M25: Semigroup rings, multiplicative semigroups of rings 13N10: Rings of differential operators and their modules
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 1 , January 2011 , pp. 284 - 318
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Adolphson, A., Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), 269–290.CrossRefGoogle Scholar
[2]Batyrev, V. V. and van Straten, D., Generalized hypergeometric functions and rational curves on Calabi–Yau complete intersections in toric varieties, Comm. Math. Phys. 168 (1995), 493–533; English summary.CrossRefGoogle Scholar
[3]Bruns, W. and Herzog, J., Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39 (Cambridge University Press, Cambridge, 1993).Google Scholar
[4]Cattani, E., D’Andrea, C. and Dickenstein, A., The A-hypergeometric system associated with a monomial curve, Duke Math. J. 99 (1999), 179–207.CrossRefGoogle Scholar
[5]Dickenstein, A., Matusevich, L. F. and Miller, E., Binomial D-modules, Duke Math. J. 151 (2010), 385–429.CrossRefGoogle Scholar
[6]Gel′fand, I. M., Graev, M. I. and Zelevinskiĭ, A. V., Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR 295 (1987), 14–19.Google Scholar
[7]Gel′fand, I. M., Zelevinskiĭ, A. V. and Kapranov, M. M., Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23 (1989), 12–26; Correction in 27 (1993), 91.Google Scholar
[8]Hochster, M., Rings of invariants of tori, Cohen–Macaulay rings generated by monomials, and polytopes, Ann. of Math. (2) 96 (1972), 318–337.CrossRefGoogle Scholar
[9]Hosono, S., Lian, B. H. and Yau, S. T., GKZ-generalized hypergeometric systems in mirror symmetry of Calabi–Yau hypersurfaces, Comm. Math. Phys. 182 (1996), 535–577.CrossRefGoogle Scholar
[10]Jow, S.-Y. and Miller, E., Multiplier ideals of sums via cellular resolutions, Math. Res. Lett. 15 (2008), 359–373.CrossRefGoogle Scholar
[11]Kashiwara, M., Systems of microdifferential equations (Birkhäuser Boston, Inc., Boston, 1983).Google Scholar
[12]Grayson, D. R. and Stillman, M. E., Macaulay 2, a software system for research in algebraic geometry, http://www.math.uiuc.edu/Macaulay2/.Google Scholar
[13]Matusevich, L. F. and Miller, E., Combinatorics of rank jumps in simplicial hypergeometric systems, Proc. Amer. Math. Soc. 134 (2005), 1375–1381.CrossRefGoogle Scholar
[14]Matusevich, L. F., Miller, E. and Walther, U., Homological methods for hypergeometric families, J. Amer. Math. Soc. 18 (2005), 919–941.CrossRefGoogle Scholar
[15]Miller, E., Cohen–Macaulay quotients of normal semigroup rings via irreducible resolutions, Math. Res. Lett. 9 (2002), 117–128.CrossRefGoogle Scholar
[16]Miller, E., Topological Cohen–Macaulay criteria for monomial ideals, in Combinatorial aspects of commutative algebra, Mangalia, Romania, 2008, eds V. Ene and E. Miller, Contemporary Mathematics, vol. 502 (American Mathematical Society, Providence, RI, 2009), 137–156.Google Scholar
[17]Miller, E. and Sturmfels, B., Combinatorial commutative algebra, Graduate Texts in Mathematics, vol. 227 (Springer, New York, 2005).Google Scholar
[18]Okounkov, A., Generating functions for intersection numbers on moduli spaces of curves, Int. Math. Res. Not. 18 (2002), 933–957.CrossRefGoogle Scholar
[19]Okuyama, G., A-hypergeometric ranks for toric threefolds, Int. Math. Res. Not. 38 (2006), Article ID 70814.Google Scholar
[20]Saito, M., Isomorphism classes of A-hypergeometric systems, Compositio Math. 128 (2001), 323–338.CrossRefGoogle Scholar
[21]Saito, M., Logarithm-free A-hypergeometric series, Duke Math. J. 115 (2002), 53–73.CrossRefGoogle Scholar
[22]Saito, M., Sturmfels, B. and Takayama, N., Gröbner deformations of hypergeometric differential equations (Springer, Berlin, 2000).CrossRefGoogle Scholar
[23]Saito, M. and Traves, W. N., Differential algebras on semigroup algebras, in Symbolic computation: solving equations in algebra, geometry, and engineering, South Hadley, MA, 2000, eds E. L. Green, S. Hosten, R. C. Laubenbacher and V. A. Powers, Contemporary Mathematics, vol. 286 (American Mathematical Society, Providence, RI, 2001), 207–226.CrossRefGoogle Scholar
[24]Schulze, M. and Walther, U., Hypergeometric D-modules and twisted Gauß–Manin systems, J. Algebra 322 (2009), 3392–3409.CrossRefGoogle Scholar
[25]Sturmfels, B., Solving algebraic equations in terms of A-hypergeometric series, in Formal power series and algebraic combinatorics, Minneapolis, MN, 1996, Discrete Math. 210 (2000), 171–181.CrossRefGoogle Scholar
[26]Sturmfels, B. and Takayama, N., Gröbner bases and hypergeometric functions, in Gröbner bases and applications, Linz, 1998, eds B. Buchberger and F. Winkler, London Mathematical Society Lecture Note Series, vol. 251 (Cambridge University Press, Cambridge, 1998), 246–258.CrossRefGoogle Scholar
[27]Walther, U., Duality and monodromy reducibility of A-hypergeometric systems, Math. Ann. 338 (2007), 55–74.CrossRefGoogle Scholar