Geometry & Singularity Theory Abstracts
Multiplicities of zero-schemes in quasi-homogeneous corank-1
singularities Cn ->
(with Ton Marar and Maria Ruas)
In Singularity Theory
(ed. Bill Bruce, David Mond).
LMS Lecture Notes Series Vol. 263, CUP, 1999
We consider corank-1 map-germs from
Cn to itself. Given any stable
perturbation of such a map, there are various multi-germ
singularity types that arise, and the object is to count the
number of each type of isolated singularity that occurs. For
example, in dimension 2 there are cusp points and double-fold
points (i.e. points in the target where 2 fold lines
intersect), and in dimension 3 there are swallowtail points,
fold-cusp points (where the cupidal edge passes through a
smooth sheet on the discriminant) and triple fold points
(where 3 sheets intersect). We give a formula for these
numbers in terms of the dimensions of certain local algebras.
In the case that the original unperturbed map is
weighted-homogeneous, we give a closed formula for these
numbers, in terms of the weights and degrees of the map. This
result is applied to finding the multiplicity of each stratum
in the discriminant of an Ak singularity. [ Paper (257K)]
The Path Formulation of Bifurcation Theory
In Dynamics, Bifurcations and Symmetry
(ed. P. Chossat).
NATO ASI Series Vol. 437, Kluwer, Dordrecht, 1994.
We show how the path formulation of bifurcation theory
can be made to work, and that it is (essentially) equivalent
to the usual parametrized contact equivalence of Golubitsky
and Schaeffer. If there is more than one bifurcation
parameter present in the problem, it seems clear that the path
approach is preferable to the distinguished parameter
approach, since the tangent spaces in question are easier to
compute. [ Paper (218K)]
Deformations of maps on complete intersections, Damon's KV-equivalence and bifurcations.
(with David Mond)
(ed. J.P. Brasselet),
LMS Lecture Notes Vol. 201, CUP, 1994.
A result due to Jim Damon relates the
Ae-versal unfoldings of a map-germ f with
the KV-versal unfoldings of an associated map germ
which induces f from a stable map G (where
V is the discriminant of G). We extend this
result to the case where the source is a complete intersection
with an isolated singularity. In a similar vein, we also
relate the bifurcation theoretic versal deformation of a
bifurcation problem (map-germ) g to the KV-versal
deformation of an associated map germ which induces g
from a versal deformation of the organizing centre
g0 of g, where this time
V is the bifurcation set of this versal deformation.
The extension of Damon's theorem is used to provide an
extension (again to cases where the source is an icis) of a
result of Damon and Mond relating the discriminant Milnor
number of a map to its left-right-codimension. [ Paper (222K)]
Multiplicities of Critical Points of Invariant Functions.
(Workshop on Real and Complex Singularites) (1993), 93-136.
The purpose of this expository article is to describe in
an elementary and homogeneous manner, the relationship between
the geometric and algebraic multiplicities of isolated
critical points of holomorphic functions. In particular, I am
interested in the setting where the function is invariant
under some group action. The emphasis is on functions
invariant under actions of finite groups as very little is
known if the group is not finite.
Most of the results
described here are already explicitly in the literature; the
only small extension is to functions that are not invariant,
but equivariant under the action of a group
G: a function f satisfying
f(gx)=char(g)f(x) for some character char:G
->C* . The results (in Section 7) on the
multiplicity of critical points of homogeneous functions
invariant under C* are also new. [ Paper (226K)]
Quotient spaces and critical points of invariant functions for C*-actions.
(with Duco van Straten)
J. reine angew. Math. 437
Consider a function on Cn
which is invariant under a linear C*
action. If it has a degenerate critical point, then under
pertubation (to another invariant function) the critical point
will break up into a certain number of group orbits of
non-degenerate critical points. How many? To answer that
question we study the fundamental algebraic properties of the
sheaves of invariant and basic differential forms for
C* actions, in order to show that the module used to
define the algebraic multiplicity is Cohen-Macaulay, so that
its dimension gives a reasonable definition of geometric
multiplicity. [ Paper
On generic composites of mappings.
Bull. L.M.S. 23
It is shown that (in the nice dimensions) if a stable map is restricted to a generic
submanifold of the source manifold, then the reulting map will also be stable. At the same
time, the analgous result for versal unfoldings is proved.
A few examples of applications to the extrinsic geometry of submanifolds are discussed.
Paper (via DOI)
One-forms on singular curves and the topology of real curve singularities.
(with Duco van Straten) Topology 29
Let C be a real analytic curve in
Rn and suppose that p is a
singular point of the curve. How does one find the number of
branches passing through the singular point?
In the case
that C is a complete intersection in some neighbourhood of p
(i.e. it is given locally by n-1 equations) then there is a
solution to this problem due to K. Aoki, T. Fukuda and
T. Nishimura. In this paper, we give a general solution to
This solution uses meromorphic differential
forms on the complexification of the curve, and properties of
the module of Rosenlicht differentials, defined in terms of
residues. A 1-form on a real curve defines an orientation, and
the method allows one to find the number of branches counted
with orientation for any given 1-form; particular choices of
the form will then give the total number of branches.
Paper (via DOI)
Surfaces in 3-space and their contact with circles.
J. Differential Geometry 23
The higher-order local extrinsic differential geometry of surfaces in 3-space is described in terms of the contact between the surface and all possible circles.
Paper (via Project Euclid)
On contact between submanifolds.
Michigan Math. J. 33
We make use of singularity theory to define the contact between submanifolds of a manifold, whether or not the submanifolds are of complementary dimension.
Paper (via DOI)