Hypersurface singularities in positive characteristic

Abstract

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; and the Taylor development involves higher order differentials. On the other hand, the multiplicity of a hypersurface can be expressed in terms of general projections defined at étale neighborhoods of such point: Given a local embedding of a ( d − 1 ) -dimensional hypersurface in a smooth d-dimensional scheme, a general projection on a smooth ( d − 1 ) -dimensional scheme is a finite map on the hypersurface; and the multiplicity at the point is the degree of this finite map. We relate both approaches. In fact we study invariants of embedded hypersurfaces, defined in terms of differential operators, which express properties of the ramification of the morphism. Of particular interest is the case of hypersurfaces over fields of positive characteristic. A central result in multiplicity theory of hypersurfaces 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. This is a key point in the proof of embedded desingularization. In this paper we provide a characteristic free approach to this form of elimination. 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, than it does in characteristic zero, when it comes to compatibility of elimination with permissible monoidal transformation. This opens the way to new questions on resolution problems. We also discuss the behavior of invariants, attached to a singularity at a point, with this form of elimination.