We consider bilevel programming problems in the so-called optimistic formulation. The aim is to study the generic structure of such problems in a neighborhood of their solutions. In particular, we are interested in local reductions of the feasible set via a description by means of a finite set of smooth inequality constraints. As a by-product, this gives optimality conditions of first and second order. We prove generic nondegenerate local reductions in the case where the lower level is unconstrained and one dimensional. Moreover, the validity in higher dimensions is conjectured. However, in case where inequality constraints appear in the lower-level problem, we show, by means of a stable example, that generically a weaker concept of local reduction has to be considered.