68 research outputs found
Cohen-Macaulayness and computation of Newton graded toric rings
Let be a positive semigroup in generated by , and let
be the associated semigroup ring over a field . We investigate
heredity of the Cohen-Macaulay property from to both its -Newton
graded ring and to its face rings. We show by example that neither one inherits
in general the Cohen-Macaulay property. On the positive side we show that for
every there exist generating sets for which the Newton graduation
preserves Cohen-Macaulayness. This gives an elementary proof for an important
vanishing result on -hypergeometric Euler-Koszul homology. As a tool for our
investigations we develop an algorithm to compute algorithmically the Newton
filtration on a toric ring.Comment: 20 pages, 4 figure
Irregularity of hypergeometric systems via slopes along coordinate subspaces
We study the irregularity sheaves attached to the -hypergeometric
-module introduced by Gel'fand et al., where
is pointed of full rank and
. More precisely, we investigate the slopes of this
module along coordinate subspaces.
In the process we describe the associated graded ring to a positive semigroup
ring for a filtration defined by an arbitrary weight vector on torus
equivariant generators. To this end we introduce the -umbrella, a
simplicial complex determined by and , and identify its facets with the
components of the associated graded ring.
We then establish a correspondence between the full -umbrella and the
components of the -characteristic variety of . We compute in
combinatorial terms the multiplicities of these components in the
-characteristic cycle of the associated Euler-Koszul complex, identifying
them with certain intersection multiplicities.
We deduce from this that slopes of are combinatorial,
independent of , and in one-to-one correspondence with jumps of the
-umbrella. This confirms a conjecture of Sturmfels and gives a converse
of a theorem of Hotta: is regular if and only if defines a
projective variety.Comment: 44 pages, 3 figures, choose PS or PDF to see figures, new Lemma 2.8
fills gap in previous version of Lemma 2.12, error in previous version of
Theorem 3.2 repaired by considering L-holonomic modules in Sections 3.2 and
4.
Computing homomorphisms between holonomic D-modules
Let K be a subfield of the complex numbers, and let D be the Weyl algebra of
K-linear differential operators on K[x_1,...,x_n]. If M and N are holonomic
left D-modules we present an algorithm that computes explicit generators for
the finite dimensional vector space hom_D(M,N). This enables us to answer
algorithmically whether two given holonomic modules are isomorphic. More
generally, our algorithm can be used to get explicit generators for
ext^i_D(M,N) for any i.Comment: 30 pages, AMS-LaTex, uses
verbatim,amsmath,latexsym,amssymb,amsbsy,diagram
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