| Let X(,0) be a germ of an n-dimensional hypersurface with isolated singularity. When n = 2, there are invariants (phi)(X(,0)) and (kappa)(X(,0)), called the number of vanishing cusps and nodes, respectively, defined by taking generic projections from a versal deformation of X(,0). The first chapter of this thesis generalizes these invariants to higher-dimensional hypersurfaces using the contact classes of Mather. If i:X(,0) (C('n+1),0) is an embedding of X(,0), we obtain invariants of the pair (X(,0),i) which are conjecturally invariants of X(,0). There is one invariant for each 0-dimensional contact class relative to (n,d), 1 (LESSTHEQ) d (LESSTHEQ) n, provided that (n,d) is good in the sense of Mather. We show that these invariants are upper-semicontinuous in families.;In chapter two of this thesis we refine a technique due to Gusein-Zade and A'Campo for calculating the intersection pairing of the Milnor fiber F of a germ of a plane curve singularity. We prove the following result, claimed but not proved by A'Campo: there is a basis for H(,1)(F;Z) such that the matrix of the intersection form relative to this basis has only 0'('s) and (+OR-)1'('s) as entries. This gives a restriction on the possible local embedded topological types of plane curve singularities. |
