540 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 $H$ be a positive semigroup in $\mathbb{Z}^d$ generated by $A$, and let
$K[H]$ be the associated semigroup ring over a field $K$. We investigate
heredity of the Cohen-Macaulay property from $K[H]$ to both its $A$-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 $H$ there exist generating sets $A$ for which the Newton graduation
preserves Cohen-Macaulayness. This gives an elementary proof for an important
vanishing result on $A$-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 $A$-hypergeometric
$D$-module $M_A(\beta)$ introduced by Gel'fand et al., where
$A\in\mathbb{Z}^{d\times n}$ is pointed of full rank and
$\beta\in\mathbb{C}^d$. 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 $L$ on torus
equivariant generators. To this end we introduce the $(A,L)$-umbrella, a
simplicial complex determined by $A$ and $L$, and identify its facets with the
components of the associated graded ring.
We then establish a correspondence between the full $(A,L)$-umbrella and the
components of the $L$-characteristic variety of $M_A(\beta)$. We compute in
combinatorial terms the multiplicities of these components in the
$L$-characteristic cycle of the associated Euler-Koszul complex, identifying
them with certain intersection multiplicities.
We deduce from this that slopes of $M_A(\beta)$ are combinatorial,
independent of $\beta$, and in one-to-one correspondence with jumps of the
$(A,L)$-umbrella. This confirms a conjecture of Sturmfels and gives a converse
of a theorem of Hotta: $M_A(\beta)$ is regular if and only if $A$ 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

### On the formal structure of logarithmic vector fields

In this article, we prove that a free divisor in a three dimensional complex
manifold must be Euler homogeneous in a strong sense if the cohomology of its
complement is the hypercohomology of its logarithmic differential forms. F.J.
Calderon-Moreno et al. conjectured this implication in all dimensions and
proved it in dimension two. We prove a theorem which describes in all
dimensions a special minimal system of generators for the module of formal
logarithmic vector fields. This formal structure theorem is closely related to
the formal decomposition of a vector field by Kyoji Saito and is used in the
proof of the above result. Another consequence of the formal structure theorem
is that the truncated Lie algebras of logarithmic vector fields up to dimension
three are solvable. We give an example that this may fail in higher dimensions.Comment: 13 page

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