374 research outputs found
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
Derivations of negative degree on quasihomogeneous isolated complete intersection singularities
J. Wahl conjectured that every quasihomogeneous isolated normal singularity
admits a positive grading for which there are no derivations of negative
weighted degree. We confirm his conjecture for quasihomogeneous isolated
complete intersection singularities of either order at least 3 or embedding
dimension at most 5. For each embedding dimension larger than 5 (and each
dimension larger than 3), we give a counter-example to Wahl's conjecture.Comment: 11 page
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
Partial normalizations of coxeter arrangements and discriminants
We study natural partial normalization spaces of Coxeter arrangements and discriminants
and relate their geometry to representation theory. The underlying ring structures arise from Dubrovin’s
Frobenius manifold structure which is lifted (without unit) to the space of the arrangement. We also
describe an independent approach to these structures via duality of maximal Cohen–Macaulay fractional
ideals. In the process, we find 3rd order differential relations for the basic invariants of the Coxeter
group. Finally, we show that our partial normalizations give rise to new free divisors
Gevrey expansions of hypergeometric integrals II
We study integral representations of the Gevrey series solutions of irregular
hypergeometric systems under certain assumptions. We prove that, for such
systems, any Gevrey series solution, along a coordinate hyperplane of its
singular support, is the asymptotic expansion of a holomorphic solution given
by a carefully chosen integral representation.Comment: 27 pages, 2 figure
Quasihomogeneity of curves and the jacobian endomorphism ring.
We give a quasihomogeneity criterion for Gorenstein curves. For complete intersections, it is related to the fi rst step of Vasconcelos' normalization algorithm. In the process, we give a simpli ed proof of the Kunz-Ruppert criterion
Dual logarithmic residues and free complete intersections
This preprint is the same as a preprint with the same title in Arxiv . org, version V3We introduce a dual logarithmic residue map for hypersurface singularities and use it to answer a question of Kyoji Saito. Our result extends a theorem of Lê and Saito by an algebraic characterization of hypersurfaces that are normal crossing in codimension one. For free divisors, we relate the latter condition to other natural conditions involving the Jacobian ideal and the normalization. This leads to an algebraic characterization of normal crossing divisors. We suggest a generalization of the notions of logarithmic vector fields and freeness for complete intersections. In the case of quasihomogeneous complete intersection space curves, we give an explicit description
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