Deformations of hypersurfaces with a one-dimensional singular locus

Abstract

A germ of an analytic function f : (Cm, 0) + (C, 0) with an isolated singularity has a Morsification, that is to say, there exists an analytic family of functions f, : + C such that f, has only A, singularities, also called Morse singularities, for all s#O small enough. The number of these A, points is equal to dimC(@/Jf), where .Zr is the Jacobi ideal off. These A, points are used to construct the homotopy type of the Milnor fibre off, see [ 141. We generalize the above theory to the case that the singular locus of f-‘(O) is a curve _X. We develop a deformation theory of the pair (f,Z) and generalize the notion of a Morsification, see also [25] and [12]. The fundamental singularities besides the A, points are the so called A, and D, singularities. In a Morsification there are a finite number of A, and D, points and their total equals dimc(Z/Jf), where Z is the ideal defining C. This answers a conjecture posed by Siersma [25]. These Morsifications are used by Siersma [26] in order to construct the homotopy type of the Milnor fibre of $ In Section 1 we recall the definition of the primitive ideal j Z of Z and construct a map Zzf : Hom,,(Z/Z2, @=) + Z/l Z and derive a formula for dimZ 0) is smoothable. In Section 4 we give an upper bound for the number of D, points in a Morsification if f E Z2. This upperbound is an equality in case (2, 0) is a complete intersection or a curve in (C3,0). Q denotes the local ring of germs of analytic functions f: (C”, 0) --f C and m its maximal ideal. d denotes the local ring of germs of analytic functions F: (C” x S, 0) + C and &z its maximal ideal. Sometimes in proofs we take representatives of the germs considered, on an open neighbourhood U of 0 in Cm or Cm x S and then Q resp. 8 denote the sheaf of analytic functions on U. We let Jf be the Jacobi ideal off, it is the ideal generated by the partial derivatives off.