Homological methods for hypergeometric families

By Laura Felicia Matusevich, Ezra Miller, Uli Walther. 2004

109 citations recorded by the source corpus.

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 x n integer matrix A and a parameter \beta \in \CC^d. To do so we introduce an Euler-Koszul functor for hypergeometric families over \CC^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 is rank-jumping for H_A(\beta) if and only if \beta lies in the Zariski closure of the set of \ZZ^d-graded degrees \alpha where the local cohomology \bigoplus_{i<d}H^i_\frakm(\CC[\NN A])_\alpha of the semigroup ring \CC[\NN A] supported at its maximal graded ideal \frakm is nonzero. Consequently, H_A(\beta) has no rank-jumps over \CC^d if and only if \CC[\NN A] is Cohen-Macaulay of dimension d.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9cdf990d40ce19ea920af412c10c7bf841107809

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In Algebra and Number Theory

In Computational Theory and Mathematics

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Citations in the corpus

Irregularity of hypergeometric systems via slopes along coordinate subspaces (2006, methodology): “ in Remark 3.3. Lemma 4.1 and [Sch85, II. Prop. 1.3.1.b] yield the following statement. Lemma 4.5. CCL is additive. Our main technical tool for the study of CC(MA(β)) is the Euler–Koszul functor from [MMW05]. For a Zd-graded left D-module M and a Zd-homogeneous y ∈ M, (4.2.1) (Ei −β i) ◦y := (Ei −βi”
Logarithmic Frobenius manifolds, hypergeometric systems and quantum D-modules (2010, methodology): “= ea i and deg(∂λi) = −eai, and the Fourier-Laplace isomorphism gives rise to an injective Zn+1-graded ring homomorphism R→ Dsending wi to −∂λi. Again in order to match our notations with those from [MMW05], let us put E0 := Pm i=0 λi∂λi ∈ Dand Ek:= Pm i=1 akiλi∂λi ∈ Dfor all k= 1,...,n. Let Pbe any”
Combinatorics of binomial primary decomposition (2008, background): “In noncommutative settings, such as [MMW05, DMM07], the variables are written ∂ 1 , . . . , ∂ n , and the degree of ∂ j is usually de ﬁ ned to be − a j instead of a j .”
Decompositions of commutative monoid congruences and binomial ideals (2011, background): “In fact, it was in the hypergeometric framework that the combinatorics of binomial primary decomposition had its origin [Matusevich et al. 2005; Dickenstein et al. 2010a; 2010b], providing tight control over series solutions.”