Hypersurface Singularities in Positive Characteristic

We present results on multiplicity theory. Differential operators on smooth schemes play a central role in the study of the multiplicity of an embedded hypersurface at a point. This follows from the fact that the multiplicity is defined by the Taylor development of the defining equation at such point. On the other hand, the multiplicity of a hypersurface at a point can be expressed in terms of general projections defined at suitablé etale neighborhoods of such point: The restriction to a hypersurface embedded in a d-dimensional smooth scheme, of a general projection on a d − 1 dimensional smooth scheme, induces a finite morphism on the hypersurface. And the multiplicity of the hypersurface at a point is also defined as the degree of this finite morphisms. In this paper we relate both approaches. In fact we study invariants of embedded hypersur-faces, defined in terms of differential operators, which express properties of the ramification of the finite morphism. Of particular interest is the case of hypersurfaces over fields of positive characteristic. A central result in multiplicity theory is a form of elimination of one variable in the description of highest multiplicity locus. This form of elimination, known over fields of characteristic zero, is achieved with the notion of Tschirnhausen polynomial introduced by Abhyankar. In this paper we provide a characteristic free approach to this form of elimination, and present new invariants. Our alternative approach is based on projections on smooth d − 1-dimensional schemes. The properties of this new form of elimination remain weaker in positive characteristic, then it does in characteristic zero, when it comes to compatibility of elimination with permissible monoidal transformation; and this opens the way to new questions. We also discuss here the behavior of other well known invariants, attached to a singularity at a point (to the tangent cone), with this form of elimination.