3,517 research outputs found
Differential forms on free and almost free divisors
We introduce a variant of the usual Kähler forms on singular free divisors, and show that it enjoys the same depth properties as Kähler forms on isolated hypersurface singularities. Using these forms it is possible to describe analytically the vanishing cohomology, and the Gauss–Manin 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
- …