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

DOI:
https://doi.org/10.1090/S0894-0347-05-00488-1

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 arising from a integer matrix and a parameter . To do so we introduce an Euler-Koszul 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 rank-jumping 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 rank-jumps over if and only if is Cohen-Macaulay of dimension .

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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:
https://doi.org/10.1090/S0894-0347-05-00488-1

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.