David Mond

Some remarks on the geometry and classification of germs of maps from surfaces to 3-space

ONE of the most striking features of the theory of singularities of analytic functions f: (Z:“, O)-+( 2.0) is the ubiquity of the Milnor number p=dim: O,/(~.?f/dz,, . . , dj/&,>. Not only is its finiteness a necessary and sufficient criterion for finite 9-determinacy, but it appears also in two geometric guises, first as the number of Morse points in a generic deformation off and second, as the rank of the middle-dimensional homology of the Milnor fibre off: If 11 is finite, then ,U + n 1 is also the codimension of theSk orbit of the k-jet off in J”(n, l), for all sufficiently large k. In the theory of singularities of maps f: (Y, O)-+(,2p, 0), with p> 1, no single number appears in such a variety of different roles. The finiteness of the de-codimension off is of course a necessary and sufficient criterion for finite d-determinacy [ 121, but the ~&e of this number does not seem to convey any geometrical information, beyond in some way reflecting the overall complexity of the singularity off: One reason for this may be sought in the fact that whereas for functions there is only one kind of stable isolated singularity, namely the Morse point, for maps 3”Z: P there may be more; the Milnor algebra 0,/(8f/Zz,, . . . , df/?z,) is just the algebra of contact of the map-germj’f: (Z-“, O)+J’(C”, C) with the manifold of nonsubmersive jets, and in other dimensions there will be more than one algebra of contact to look at. A difficulty that immediately arises here is that the isolated stable singularities may be not monobut multi-germ singularities, and since we do not know how to fill in the missing diagonal in the multi-jet space, ,J’(G”, Cp ), there is not always a natural way of obtaining a corresponding algebra of contact. However, the notion of the number of stable isolated singularities of each kind, in a generic deformation off, is well defined: in a versa1 unfolding F: (2” x Zd, O)+(Sp x Zd, 0), the bifurcation set B G Zd is a proper subvariety and so does not separate Zd; thus, for any two points, u, c in Zd -B, the generic deformationsf, andf, of fmay be joined by a curve of generic deformations, and hence must have the same number of singularities of each kind. In this paper we consider germs (Z”, O)+(:P, 0) with p > n, of corank 1, and in particular the case n = 2, p = 3; here there are two isolated stable singularities, the cross-cap (Whitney umbrella, or pinch point) and the triple point. Under the hypothesis that the rank offat 0 is 1, we show how to count the number of cross-caps C(J) in a generic deformation off(in

\(A\) and the vanishing topology of discriminants

2) and the number of triple points r(J) (in 53). It is clear that forfto be finitely determined one must have both C(J) and r(J) finite, but this is not enough, and in

Stochastic factorizations, sandwiched simplices and the topology of the space of explanations

4 we introduce a new integervalued invariant NY-), whose finiteness, together with that of C(J) and r(J), is equivalent to finite &-determinancy (Theorem 4.7). N(J) measures, in some sense, the non-transverse selfintersection concentrated at the origin.

Tjurina and Milnor Numbers of Matrix Singularities

SummarySuppose thatf: ℂn, 0→ℂp, 0 is finitely

Linear free divisors and Frobenius manifolds

A

\Bbb R

-determined withn≧p. We define a “Milnor fiber” for the discriminant off; it is the discriminant of a “stabilization” off. We prove that this “discriminant Milnor fiber” has the homotopy type of a wedge of spheres of dimensionp−1, whose number we denote byµΔ(f). One of the main theorems of the paper is a “μ=τ” type result: if (n, p) is in the range of nice dimensions in the sense of Mather, then

\Bbb C

\mu _\Delta (f) \geqq A_e