In this article, we investigate the geometry of quasi homogeneous co-rank one finitely determined map germs from to  with  We give a complete description, in terms of the weights and degrees, of the invariants that are associated to all stable singularities which appear in the discriminant of such map germs. The first class of invariants which we study are the isolated singularities, called 0-stable singularities because they are the 0-dimensional singularities. First, we give a formula to compute the number of  points which appear in any stable deformation of a quasi homogeneous co-rank one map germ from  to  with  To get such a formula, we apply the Hilbert’s syzygy theorem to determine the graded free resolution given by the syzygy modules of the associated iterated Jacobian ideal. Then we show how to obtain the other 0-stable singularities, these isolated singularities are formed by multiple points and here we use the relation among them and the Fitting ideals of the discriminant. For  there exists only the germ of double points set and for  there are the triple points, named points and the normal crossing between a germ of a cuspidal edge and a germ of a plane, named  For  there appear also the one-dimensional singularities, which are of two types: germs of cuspidal edges or germs of double points curves. For these singularities, we show how to compute the polar multiplicities and also the local Euler obstruction at the origin in terms of the weights and degrees.

geometry of quasi homogeneous map germs, invariants of stable singularities.