Irregular hypergeometric D-modules

Details Irregular hypergeometric D-modules Irregular hypergeometric D-modules

2009-06-18

arxiv.org/abs/0906.3478v2

Mathematics

Algebraic Geometry

41 pages; references, Remark 7.2. and 4 figures added; some comments changed; corrected typos

Scientific paper

We study the irregularity of hypergeometric D-modules $\mathcal{M}_A (\beta )$ via the explicit construction of Gevrey series solutions along coordinate subspaces in $X =\mathbb{C}^n$. As a consequence, we prove that along coordinate hyperplanes the combinatorial characterization of the slopes of $\mathcal{M}_A (\beta)$ given by M. Schulze and U. Walther in [21] still holds without any assumption on the matrix A. We also provide a lower bound for the dimensions of the spaces of Gevrey solutions along coordinate subspaces in terms of volumes of polytopes and prove the equality for very generic parameters. Holomorphic solutions outside the singular locus of $\mathcal{M}_A (\beta)$ can be understood as Gevrey solutions of order one along X at generic points and so they are included as a particular case.

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