137 research outputs found
Differential forms on free and almost free divisors
We introduce a variant of the usual KƤhler forms on singular free divisors, and show that it enjoys the same depth properties as KƤhler forms on isolated hypersurface singularities. Using these forms it is possible to describe analytically the vanishing cohomology, and the GaussāManin connection, in families of free divisors, in precise analogy with the classical description for the Milnor fibration of an isolated complete intersection singularity, due to Brieskorn and Greuel. This applies in particular to the family Formula of discriminants of a versal deformation Formula of a singularity of a mapping
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
Linear free divisors and quiver representations
Linear free divisors are free divisors, in the sense of K.Saito, with linear
presentation matrix (example: normal crossing divisors). Using techniques of
deformation theory on representations of quivers, we exhibit families of linear
free divisors as discriminants in representation spaces for real Schur roots of
a finite quiver. We review some basic material on quiver representations, and
explain in detail how to verify whether the discriminant is a free divisor and
how to determine its components and their equations, using techniques of A.
Schofield. As an illustration, the linear free divisors that arise as the
discriminant from the highest roots of Dynkin quivers of type E7 and E8 are
treated explicitly.Comment: 27 pages; to appear in Singularities and Computer Algebra, papers in
honour of G.-M.Greuel's 60th birthda
Milnor number equals Tjurina number for functions on space curves
The equality of the Milnor number and Tjurina number for functions on space curve singularities, as conjectured recently by V. Goryunov, is proved. As a consequence, the discriminant in such a situation is a free divisor
Tjurina and Milnor numbers of matrix singularities
To gain understanding of the deformations of determinants and Pfaffians resulting from deformations of matrices, the deformation theory of composites f ā¦ F with isolated singularities is studied, where f : Y āāC is a function with (possibly non-isolated) singularity and F : X āāY
is a map into the domain of f, and F only is deformed. The corresponding T1(F) is identified as (something like) the cohomology of a derived functor, and a canonical long exact sequence is constructed from which it follows that
Ļ = Ī¼(f ā¦ F) ā Ī²0 + Ī²1,
where Ļ is the length of T1(F) and Ī²i is the length of ToriOY(OY/Jf, OX). This explains numerical coincidences observed in lists of simple matrix singularities due to Bruce, Tari, Goryunov, Zakalyukin and Haslinger. When f has CohenāMacaulay singular locus (for example when f is the
determinant function), relations between Ļ and the rank of the vanishing homology of the zero locus of f ā¦ F are obtained
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 Dubrovinās
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 CohenāMacaulay 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
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