On the rank of an A$A$‐hypergeometric D$D$‐module versus the normalized volume of A$A$
On the rank of an A$A$‐hypergeometric D$D$‐module versus the normalized volume of A$A$
复制标题
关于 A$A$ 超几何 D$D$ 模块的等级与 A$A$ 标准化体积的关系
DOI：
10.1112/blms.12567
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发表时间：
2022
影响因子：
0.9
通讯作者：
Fernández‐Fernández, María‐Cruz
中科院分区：
文献类型：
--
作者：
Berkesch, Christine;Fernández‐Fernández, María‐Cruz
The rank of an A$A$‐hypergeometric D$D$‐module MA(β)$M_A(\beta )$, associated with a full‐rank (d×n)$(d\times n)$‐matrix A$A$ and a vector of parameters β∈Cd$\beta \in {\mathbb {C}}^d$, is known to be the normalized volume of A$A$, denoted vol(A)${\operatorname{vol}}(A)$, when β$\beta$ lies outside the exceptional arrangement E(A)${\mathcal {E}}(A)$, an affine subspace arrangement of codimension at least two. If β∈E(A)$\beta \in {\mathcal {E}}(A)$ is simple, we prove that d−1$d-1$ is a tight upper bound for the ratio rank(MA(β))/vol(A)${\operatorname{rank}}(M_A(\beta ))/{\operatorname{vol}}(A)$ for any d⩾3$d\geqslant 3$. We also prove that the set of parameters β$\beta$ such that this ratio is at least two is an affine subspace arrangement of codimension at least three.
The rank of an A$A$‐hypergeometric D$D$‐module MA(β)$M_A(\beta )$, associated with a full‐rank (d×n)$(d\times n)$‐matrix A$A$ and a vector of parameters β∈Cd$\beta \in {\mathbb {C}}^d$, is known to be the normalized volume of A$A$, denoted vol(A)${\operatorname{vol}}(A)$, when β$\beta$ lies outside the exceptional arrangement E(A)${\mathcal {E}}(A)$, an affine subspace arrangement of codimension at least two. If β∈E(A)$\beta \in {\mathcal {E}}(A)$ is simple, we prove that d−1$d-1$ is a tight upper bound for the ratio rank(MA(β))/vol(A)${\operatorname{rank}}(M_A(\beta ))/{\operatorname{vol}}(A)$ for any d⩾3$d\geqslant 3$. We also prove that the set of parameters β$\beta$ such that this ratio is at least two is an affine subspace arrangement of codimension at least three.