Uniform approximation of Betti numbers

We prove that Lefschetz's principle of approximating the cohomology of a possibly singular affine scheme of finite type over a field by the cohomology of a suitable (thickening of a) hyperplane section can be made uniform: in the affine case, we can choose the hyperplane section independently of the cohomology. Using Jouanolou's trick, this gives a new way to bound the Betti numbers of quasi-projective schemes over a field, independently of the cohomology. This is achieved through a motivic version of Deligne's generic base change formula and an axiomatic presentation of the theory of perverse sheaves. These methods produce generating families of Voevodsky motivic sheaves that are realized in perverse sheaves over any base of equal characteristic.