541 research outputs found
Freeness and multirestriction of hyperplane arrangements
Generalizing a result of Yoshinaga in dimension 3, we show that a central
hyperplane arrangement in 4-space is free exactly if its restriction with
multiplicities to a fixed hyperplane of the arrangement is free and its reduced
characteristic polynomial equals the characteristic polynomial of this
restriction. We show that the same statement holds true in any dimension when
imposing certain tameness hypotheses.Comment: 8 page
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.
Quasihomogeneity of curves and the Jacobian endomorphism ring
We give a quasihomogeneity criterion for Gorenstein curves. For complete
intersections, it is related to the first step of Vasconcelos' normalization
algorithm. In the process, we give a simplified proof of the Kunz-Ruppert
criterion.Comment: 9 page
Adjoint divisors and free divisors
We describe two situations where adding the adjoint divisor to a divisor D
with smooth normalization yields a free divisor. Both also involve stability or
versality. In the first, D is the image of a corank one stable germ of a map
from complex n-space to complex (n+1)-space, and is not free. In the second, D
is the discriminant of a versal deformation of a weighted homogeneous function
with isolated critical point (subject to certain numerical conditions on the
weights). Here D itself is already free. We also prove an elementary result,
inspired by these first two, from which we obtain a plethora of new examples of
free divisors. The presented results seem to scratch the surface of a more
general phenomenon that is still to be revealed.Comment: 24 pages, 1 figur
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