Homological methods for hypergeometric families
Authors:
Laura Felicia Matusevich, Ezra Miller and Uli Walther
Journal:
J. Amer. Math. Soc. 18 (2005), 919941
MSC (2000):
Primary 13N10, 13D45, 14D99, 13F99, 16E99; Secondary 32C38, 35A27, 14M25, 70F20, 33C70, 13C14, 13D07
Published electronically:
May 25, 2005
MathSciNet review:
2163866
Fulltext PDF Free Access
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 rankjumps in this general setting. Then we investigate rankjump behavior for hypergeometric systems arising from a integer matrix and a parameter . To do so we introduce an EulerKoszul functor for hypergeometric families over , 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 is rankjumping for if and only if lies in the Zariski closure of the set of graded degrees where the local cohomology of the semigroup ring supported at its maximal graded ideal is nonzero. Consequently, has no rankjumps over if and only if is CohenMacaulay of dimension .
 [Ado94]
Alan Adolphson, Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), no. 2, 269290. MR 1262208 (96c:33020)
 [Ado99]
, Higher solutions of hypergeometric systems and Dwork cohomology, Rend. Sem. Mat. Univ. Padova 101 (1999), 179190. MR 1705287 (2001b:14032)
 [BH93]
Winfried Bruns and Jürgen Herzog, CohenMacaulay 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, NorthHolland Mathematical Library, vol. 21, NorthHolland Publishing Co., Amsterdam, 1979. MR 0549189 (82g:32013)
 [CDD99]
Eduardo Cattani, Carlos D'Andrea, and Alicia Dickenstein, The hypergeometric system associated with a monomial curve, Duke Math. J. 99 (1999), no. 2, 179207. MR 1708034 (2001f:33018)
 [CDS01]
Eduardo Cattani, Alicia Dickenstein, and Bernd Sturmfels, Rational hypergeometric functions, Compositio Math. 128 (2001), no. 2, 217239. 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, SpringerVerlag, New York, 1995. MR 1322960 (97a:13001)
 [GGZ87]
I. M. Gelfand, M. I. Graev, and A. V. Zelevinskii, Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR 295 (1987), no. 1, 1419. MR 0902936 (88j:58118)
 [GKZ89]
I. M. Gelfand, A. V. Zelevinskii, and M. M. Kapranov, Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 1226. 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, SpringerVerlag, New York, 1977. MR 0463157 (57:3116)
 [Hot98]
Ryoshi Hotta, Equivariant modules, 1998. arXiv:math.RT/9805021
 [Mat01]
Laura Felicia Matusevich, Rank jumps in codimension 2 hypergeometric systems, J. Symbolic Comput. 32 (2001), no. 6, 619641, Effective methods in rings of differential operators. MR 1866707 (2003f:33017)
 [Mat03]
, Exceptional parameters for generic hypergeometric systems, Int. Math. Res. Not. (2003), no. 22, 12251248. 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 hypergeometric systems through Laurent polynomials, 2004. arXiv:math.CO/0404183
 [Mil02]
Ezra Miller, Graded GreenleesMay 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. 233253.MR 1888202 (2004b:13019)
 [MS04]
Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics Vol. 227, SpringerVerlag, New York, 2004. MR 2110098
 [Sai01]
Mutsumi Saito, Isomorphism classes of hypergeometric systems, Compositio Math. 128 (2001), no. 3, 323338. MR 1858340 (2003f:33019)
 [Sai02]
Mutsumi Saito, Logarithmfree hypergeometric series, Duke Math. J. 115 (2002), no. 1, 5373. 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, SpringerVerlag, 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. 246258. MR 1708882 (2001c:33026)
 [Ado94]
 Alan Adolphson, Hypergeometric functions and rings generated by monomials, Duke Math. J. 73 (1994), no. 2, 269290. MR 1262208 (96c:33020)
 [Ado99]
 , Higher solutions of hypergeometric systems and Dwork cohomology, Rend. Sem. Mat. Univ. Padova 101 (1999), 179190. MR 1705287 (2001b:14032)
 [BH93]
 Winfried Bruns and Jürgen Herzog, CohenMacaulay 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, NorthHolland Mathematical Library, vol. 21, NorthHolland Publishing Co., Amsterdam, 1979. MR 0549189 (82g:32013)
 [CDD99]
 Eduardo Cattani, Carlos D'Andrea, and Alicia Dickenstein, The hypergeometric system associated with a monomial curve, Duke Math. J. 99 (1999), no. 2, 179207. MR 1708034 (2001f:33018)
 [CDS01]
 Eduardo Cattani, Alicia Dickenstein, and Bernd Sturmfels, Rational hypergeometric functions, Compositio Math. 128 (2001), no. 2, 217239. 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, SpringerVerlag, New York, 1995. MR 1322960 (97a:13001)
 [GGZ87]
 I. M. Gelfand, M. I. Graev, and A. V. Zelevinskii, Holonomic systems of equations and series of hypergeometric type, Dokl. Akad. Nauk SSSR 295 (1987), no. 1, 1419. MR 0902936 (88j:58118)
 [GKZ89]
 I. M. Gelfand, A. V. Zelevinskii, and M. M. Kapranov, Hypergeometric functions and toric varieties, Funktsional. Anal. i Prilozhen. 23 (1989), no. 2, 1226. 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, SpringerVerlag, New York, 1977. MR 0463157 (57:3116)
 [Hot98]
 Ryoshi Hotta, Equivariant modules, 1998. arXiv:math.RT/9805021
 [Mat01]
 Laura Felicia Matusevich, Rank jumps in codimension 2 hypergeometric systems, J. Symbolic Comput. 32 (2001), no. 6, 619641, Effective methods in rings of differential operators. MR 1866707 (2003f:33017)
 [Mat03]
 , Exceptional parameters for generic hypergeometric systems, Int. Math. Res. Not. (2003), no. 22, 12251248. 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 hypergeometric systems through Laurent polynomials, 2004. arXiv:math.CO/0404183
 [Mil02]
 Ezra Miller, Graded GreenleesMay 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. 233253.MR 1888202 (2004b:13019)
 [MS04]
 Ezra Miller and Bernd Sturmfels, Combinatorial commutative algebra, Graduate Texts in Mathematics Vol. 227, SpringerVerlag, New York, 2004. MR 2110098
 [Sai01]
 Mutsumi Saito, Isomorphism classes of hypergeometric systems, Compositio Math. 128 (2001), no. 3, 323338. MR 1858340 (2003f:33019)
 [Sai02]
 Mutsumi Saito, Logarithmfree hypergeometric series, Duke Math. J. 115 (2002), no. 1, 5373. 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, SpringerVerlag, 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. 246258. 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/S0894034705004881
PII:
S 08940347(05)004881
Keywords:
Hypergeometric system,
CohenMacaulay,
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 DMS0304789
The third author was partially supported by the DfG, the Humboldt foundation, and NSF Grant DMS0100509
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.
