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Elimination algebras and inductive arguments in resolution of singularities

Over fields of characteristic zero, resolution of singularities is achieved by means of an inductive argument, which is sustained on the existence of the so called hypersurfaces of maximal contact. We report here on an alternative approach which replaces hypersurfaces of maximal contact by generic projections. Projections can be defined in arbitrary characteristic, and this approach has led to new invariants when applied to the open problem of resolution of singularities over arbitrary fields. We show here how projections lead to a form of elimination of one variable using invariants that, to some extent, generalize the notion of discriminant. This exposition draws special attention on this form of elimination, on its motivation, and its use as an alternative approach to inductive arguments in resolution of singularities. Using techniques of projections and elimination one can also recover some well known results. We illustrate this by showing that the Hilbert-Samuel stratum of a d-dimensional non-smooth variety can be described with equations involving at most d variables. In addition this alternative approach, when applied over fields of characteristic zero, provides a conceptual simplification of the theorem of resolution of singularities as it trivializes the globalization of local invariants.