Flache T¹-Stabilität in der Strati-fizierung verseller Entfaltungen

Abstract

Let 0 be the local ring of a simple singularity defined over the complex numbers and τ the dimension of its versal deformation space. Than it is well known that any nearby singularity in this space is also simple and has smaller unfolding dimension in the hierarchy of simple singularities. In particular this implies that the τ=max-stratum consists just of one point namely the given singularity. We want to generalize this concept as we are interested in families of varieties with formal unchanged singularities. For this we introduce in quite generality the notion of flat T¹-stabi1ity which may be checked for any k- algebra 0 where k is for simplicity an algebraically closed field of à priori arbitrary characteristics. We call 0 formal flat T¹ stable or for short flat T¹-stable if the following is true: if R is any deformation of 0 over an Artin local finite k-algebra A and if T¹(R/A,R) is A-flat than R is isomorphic to the trivial deformation$0\mathop \otimes \limits_K A$. T¹(R/A,R) is the first cotangent module of R over A with values in R. Obviously the simple singularities A_k, D_k, E₆, E₇, E₈ fulfill this criterion over C but we look also at fibres of arbitrary stable map germs, generic singularities of algebraic varieties where we have to modify this notion in order to deal with wild ramification and to quasihomo-genous hypersurface singularities where it functorializes because in this case T¹ commutes with arbitrary base change. The notion of flat T¹-stable singularities is closely related to questions of existence of equisingular families and is used in[12] and [5], [6] to stratify certain Hilbert schemes.