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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Homological methods for hypergeometric families


Authors: Laura Felicia Matusevich, Ezra Miller and Uli Walther
Journal: J. Amer. Math. Soc. 18 (2005), 919-941
MSC (2000): Primary 13N10, 13D45, 14D99, 13F99, 16E99; Secondary 32C38, 35A27, 14M25, 70F20, 33C70, 13C14, 13D07
Published electronically: May 25, 2005
MathSciNet review: 2163866
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Abstract | References | Similar Articles | Additional Information

Abstract: We analyze the behavior of the holonomic rank in families of holonomic systems over complex algebraic varieties by providing homological criteria for rank-jumps in this general setting. Then we investigate rank-jump behavior for hypergeometric systems  $H_A(\beta)$ arising from a $d \times n$ integer matrix $A$ and a parameter $\beta \in \mathbb{C} ^d$. To do so we introduce an Euler-Koszul functor for hypergeometric families over  $\mathbb{C} ^d$, whose homology generalizes the notion of a hypergeometric system, and we prove a homology isomorphism with our general homological construction above. We show that a parameter $\beta \in \mathbb{C} ^d$ is rank-jumping for $H_A(\beta)$ if and only if $\beta$ lies in the Zariski closure of the set of $\mathbb{C} ^d$-graded degrees $\alpha$ where the local cohomology $\bigoplus_{i < d} H^i_\mathfrak m(\mathbb{C} [\mathbb{N} A])_\alpha$of the semigroup ring $\mathbb{C} [\mathbb{N} A]$ supported at its maximal graded ideal  $\mathfrak m$ is nonzero. Consequently, $H_A(\beta)$ has no rank-jumps over  $\mathbb{C} ^d$ if and only if $\mathbb{C} [\mathbb{N} A]$ is Cohen-Macaulay of dimension $d$.


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  • [Ado94] Alan Adolphson, Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), no. 2, 269-290. MR 1262208 (96c:33020)
  • [Ado99] -, Higher solutions of hypergeometric systems and Dwork cohomology, Rend. Sem. Mat. Univ. Padova 101 (1999), 179-190. MR 1705287 (2001b:14032)
  • [BH93] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956 (95h:13020)
  • [Bjö79] J.-E. Björk, Rings of differential operators, North-Holland Mathematical Library, vol. 21, North-Holland Publishing Co., Amsterdam, 1979. MR 0549189 (82g:32013)
  • [CDD99] Eduardo Cattani, Carlos D'Andrea, and Alicia Dickenstein, The ${A}$-hypergeometric system associated with a monomial curve, Duke Math. J. 99 (1999), no. 2, 179-207. MR 1708034 (2001f:33018)
  • [CDS01] Eduardo Cattani, Alicia Dickenstein, and Bernd Sturmfels, Rational hypergeometric functions, Compositio Math. 128 (2001), no. 2, 217-239. MR 1850183 (2003f:33016)
  • [CK99] David Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, Amer. Math. Soc., Providence, RI, 1999. MR 1677117 (2000d:14048)
  • [Eis95] David Eisenbud, Commutative algebra, with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. MR 1322960 (97a:13001)
  • [GGZ87] I. M. Gel$'$fand, M. I. Graev, and A. V. Zelevinskii, Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR 295 (1987), no. 1, 14-19. MR 0902936 (88j:58118)
  • [GKZ89] I. M. Gel$'$fand, A. V. Zelevinskii, and M. M. Kapranov, Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 12-26. Correction in ibid, 27 (1993), no. 4, 91. MR 1011353 (90m:22025), MR 1264328 (95a:22010)
  • [Har77] Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, vol. 52, Springer-Verlag, New York, 1977. MR 0463157 (57:3116)
  • [Hot98] Ryoshi Hotta, Equivariant ${D}$-modules, 1998. arXiv:math.RT/9805021
  • [Mat01] Laura Felicia Matusevich, Rank jumps in codimension 2 $A$-hypergeometric systems, J. Symbolic Comput. 32 (2001), no. 6, 619-641, Effective methods in rings of differential operators. MR 1866707 (2003f:33017)
  • [Mat03] -, Exceptional parameters for generic $A$-hypergeometric systems, Int. Math. Res. Not. (2003), no. 22, 1225-1248. MR 1967406 (2004b:16039)
  • [MM05] Laura Felicia Matusevich and Ezra Miller, Combinatorics of rank jumps in simplicial hypergeometric systems, Proc. Amer. Math. Soc., to appear, 2005. arXiv:math.AC/0402071
  • [MW04] Laura Felicia Matusevich and Uli Walther, Arbitrary rank jumps for $A$-hypergeometric systems through Laurent polynomials, 2004. arXiv:math.CO/0404183
  • [Mil02] Ezra Miller, Graded Greenlees-May duality and the Cech hull, Local cohomology and its applications (Guanajuato, 1999), Lecture Notes in Pure and Appl. Math., vol. 226, Dekker, New York, 2002, pp. 233-253.MR 1888202 (2004b:13019)
  • [MS04] Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics Vol. 227, Springer-Verlag, New York, 2004. MR 2110098
  • [Sai01] Mutsumi Saito, Isomorphism classes of $A$-hypergeometric systems, Compositio Math. 128 (2001), no. 3, 323-338. MR 1858340 (2003f:33019)
  • [Sai02] Mutsumi Saito, Logarithm-free $A$-hypergeometric series, Duke Math. J. 115 (2002), no. 1, 53-73. MR 1932325 (2004f:16041)
  • [SST00] Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama, Gröbner deformations of hypergeometric differential equations, Algorithms and Computation in Mathematics, vol. 6, Springer-Verlag, Berlin, 2000. MR 1734566 (2001i:13036)
  • [ST98] Bernd Sturmfels and Nobuki Takayama, Gröbner bases and hypergeometric functions, Gröbner bases and applications (Linz, 1998), London Math. Soc. Lecture Note Ser., vol. 251, Cambridge Univ. Press, Cambridge, 1998, pp. 246-258. MR 1708882 (2001c:33026)

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Additional Information

Laura Felicia Matusevich
Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Address at time of publication: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Email: lfm@math.upenn.edu

Ezra Miller
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: ezra@math.umn.edu

Uli Walther
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: walther@math.purdue.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-05-00488-1
PII: S 0894-0347(05)00488-1
Keywords: Hypergeometric system, Cohen--Macaulay, toric, local cohomology, holonomic, $D$-module
Received by editor(s): June 22, 2004
Published electronically: May 25, 2005
Additional Notes: The first author was partially supported by a postdoctoral fellowship from MSRI and an NSF Postdoctoral Fellowship
The second author was partially supported by NSF Grant DMS-0304789
The third author was partially supported by the DfG, the Humboldt foundation, and NSF Grant DMS-0100509
Dedicated: Uli Walther dedicates this paper to the memory of his father, Hansjoachim Walther.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.