28 research outputs found
Characteristic cycles and Gevrey series solutions of -hypergeometric systems
We compute the -characteristic cycle of an -hypergeometric system and
higher Euler-Koszul homology modules of the toric ring. We also prove upper
semicontinuity results about the multiplicities in these cycles and apply our
results to analyze the behavior of Gevrey solution spaces of the system.Comment: 22 page
An Array of Disjoint Maximal Constant Weight Codes
We show that when gcd(n,w) = 1, the set of binary words of length n and weight w can be partitioned to give n maximal w-weight codes. It follows that under the same hypothesis, the least cardinal of a maximal constant weight code is at most 1/n times n choose w
Poset structures in Boij-S\"oderberg theory
Boij-S\"oderberg theory is the study of two cones: the cone of cohomology
tables of coherent sheaves over projective space and the cone of standard
graded minimal free resolutions over a polynomial ring. Each cone has a
simplicial fan structure induced by a partial order on its extremal rays. We
provide a new interpretation of these partial orders in terms of the existence
of nonzero homomorphisms, for both the general and the equivariant
constructions. These results provide new insights into the families of sheaves
and modules at the heart of Boij-S\"oderberg theory: supernatural sheaves and
Cohen-Macaulay modules with pure resolutions. In addition, our results strongly
suggest the naturality of these partial orders, and they provide tools for
extending Boij-S\"oderberg theory to other graded rings and projective
varieties.Comment: 23 pages; v2: Added Section 8, reordered previous section
Systems of parameters and holonomicity of A-hypergeometric systems
The main result is an elementary proof of holonomicity for A-hypergeometric
systems, with no requirements on the behavior of their singularities,
originally due to Adolphson [Ado94] after the regular singular case by Gelfand
and Gelfand [GG86]. Our method yields a direct de novo proof that
A-hypergeometric systems form holonomic families over their parameter spaces,
as shown by Matusevich, Miller, and Walther [MMW05]
A-graded methods for monomial ideals
We use \ZZ^d-gradings to study d-dimensional monomial ideals. The Koszul
functor is employed to interpret the quasidegrees of local cohomology in terms
of the geometry of distractions and to explicitly compute the multiplicities of
exponents. These multigraded techniques originate from the study of
hypergeometric systems of differential equations.Comment: Reorganized version with new introduction, Section 2 simplified,
corrections made to Section