Axiomatic theory for transversality and Bertini type theorems

Let k be a field of any characteristic and not necessarily algebraically closed. In this paper we extend some results of the transversality theory to larger families of k-schemes and of k-morphisms and to all the local geometric properties satisfying some given axioms (for example: geom, regular, geom. reduced, geom. normal and the properties S-r and R-s). As an application, we obtain a generalization of the well known Bertini type theorems for local geometric axiomatic properties.