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 ( β ) H_A(\beta ) arising from a d × n d \times n integer matrix A A and a parameter β ∈ C d \beta \in \mathbb {C}^d . To do so we introduce an Euler–Koszul functor for hypergeometric families over C d \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 β ∈ C d \beta \in \mathbb {C}^d is rank-jumping for H A ( β ) H_A(\beta ) if and only if β \beta lies in the Zariski closure of the set of C d \mathbb {C}^d -graded degrees α \alpha where the local cohomology ⨁ i > d H m i ( C [ N A ] ) α \bigoplus _{i > d} H^i_\mathfrak m(\mathbb {C}[\mathbb {N} A])_\alpha of the semigroup ring C [ N A ] \mathbb {C}[\mathbb {N} A] supported at its maximal graded ideal m \mathfrak m is nonzero. Consequently, H A ( β ) H_A(\beta ) has no rank-jumps over C d \mathbb {C}^d if and only if C [ N A ] \mathbb {C}[\mathbb {N} A] is Cohen–Macaulay of dimension d d .
We study quotients of the Weyl algebra by left ideals whose generators consist of an arbitrary Z d -graded binomial ideal I in C[∂ 1 , . . . , ∂ n ] along with Euler operators defined by the grading and a parameter β ∈ C d . We determine the parameters β for which these D-modules (i) are holonomic (equivalently, regular holonomic, when I is standard-graded); (ii) decompose as direct sums indexed by the primary components of I; and (iii) have holonomic rank greater than the rank for generic β. In each of these three cases, the parameters in question are precisely those outside of a certain explicitly described affine subspace arrangement in C d . In the special case of Horn hypergeometric D-modules, when I is a lattice basis ideal, we furthermore compute the generic holonomic rank combinatorially and write down a basis of solutions in terms of associated A-hypergeometric functions. This study relies fundamentally on the explicit lattice point description of the primary components of an arbitrary binomial ideal in characteristic zero, which we derive in our companion article [DMM08]. CONTENTSTo construct a binomial D-module, the starting point is an integer matrix A, about which we wish to be consistent throughout.Convention 1.2. A = (a ij ) ∈ Z d×n denotes an integer d×n matrix of rank d whose columns a 1 , . . . , a n all lie in a single open linear half-space of R d ; equivalently, the cone generated by the columns of A is pointed (contains no lines), and all of the a i are nonzero. We also assume that ZA = Z d ; that is, the columns of A span Z d as a lattice.The reformulation of Horn systems in Section 1.4 proceeds by a change of variables, so we will use x = x 1 , . . . , x n and ∂ = ∂ 1 , . . . , ∂ n (where ∂ i = ∂ x i = ∂/∂x i ), instead of z 1 , . . . , z m and ∂ z 1 , . . . , ∂ zm , whenever we work in the binomial setting. The matrix A induces a Z dgrading of the polynomial ring C[∂ 1 , . . . , ∂ n ] = C[∂], which we call the A-grading, byis A-graded if it is generated by elements that are homogeneous for the A-grading. For example, a binomial ideal is generated by binomials ∂ u −λ∂ v , where u, v ∈ Z n are column vectors and λ ∈ C; such an ideal is A-graded precisely when it is generated by binomials ∂ u − λ∂ v each of which satisfies either Au = Av or λ = 0 (in particular, monomials are allowed as generators of binomial ideals). The hypotheses on A mean that the A-grading is a positive Z d -grading [MS05, Chapter 8].The Weyl algebra D = D n of linear partial differential operators, written with the variables x and ∂, is also naturally A-graded by additionally setting deg(x i ) = a i . Consequently, the Euler operators in our next definition are A-homogeneous of degree 0. Definition 1.3. For each i ∈ {1, . . . , d}, the i th Euler operator isGiven a vector β ∈ C d , we write E − β for the sequence E 1 − β 1 , . . . , E d − β d . (The dependence of the Euler operators E i on the matrix A is suppressed from the notation.) Lemma 7.10. If β is A J -nonresonant, then for any γ ∈ N J , and for all torus trans...
We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex holomorphic solutions. We also obtain analogous results for the generalized hypergeometric systems arising from lattices of any rank.
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