Generic Gröbner Bases: Application to the Constructibility of the Set of the Algebraic and Analytic Gröbner Fans

The contribution of this paper lies in two aspects. The first one deals with a natural notion of generic Gröbner (or standard) bases on an irreducible affine scheme for an ideal depending on parameters. This takes place in rings of differential operators and concerns the algebraic and the formal case. This gives a generalization of some results of V. Weispfenning obtained in a commutative and algebraic situation. The second aspect is an application of the first one to the study of the behaviour of the algebraic or analytic Gröbner fan of a monogeneous D-module depending on parameters. In substance, we prove that the Gröbner fan is generically constant.