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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: 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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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.