On the middle Betti number of certain singularities with critical locus a hyperplane

Abstract

We study holomorphic germs $f : (\mathbb{C}^{n+1}, 0) \to (\mathbb{C}, 0)$ with the following properties:

(i) the critical set $H$ of the germ $f$ is a hyperplane $H = \{x = 0\}$;

(ii) the transversal singularity of the germ $f$ in points of the set $H \setminus \{0\}$ has type $A_{k-1}$.

We will investigate the topological structure of the Milnor fibre for $f$ and give explicit formula for the middle Betti number of the Milnor fibre and for the quasihomogeneous case we express it in terms of weights and degrees.