\(\mathcal A\)-codimension and the vanishing topology of discriminants
Let \(f:(\mathbb{C}^ n,0)\to(\mathbb{C}^ 0,0)\) be finitely \({\mathcal A}\)- determined, \(n\geq p\). If \((n,p)\) is in a range of nice dimensions in the sense of Mather then there is a deformation \(f_ t\) which has a stable representative on a neighborhood \(U\) of \(O\) called stabilization of \(f\). It is proved that the Milnor fibre of the discriminant of \(f_ t\) has the homotopy type of a wedge of \(\mu_ \Delta(f)\) spheres of dimensions \(p-1\cdot\mu_ \Delta(f)\) is called the discriminant Milnor number of \(f\). It is proved that \(\mu_ \Delta(f)=A_ e\)-codim \((f)\) and equality holds if \(f\) is weighted homogeneous (a result similar to \(\mu(f)=\tau(f)\) for singularities). If \((n,p)\) is not in a range of nice dimensions correction terms for the inequality are given.
scientific article; zbMATH DE number 59800
scientific article