Legendrian varieties and quasi-ordinary hypersurfaces

Abstract

This thesis is a study of the Legendrian Varieties that are conormals of quasi-ordinary hypersurfaces. In the first chapter we study the analytic classification of the Legendrian curves that are the conormal of a plane curve with a single Puiseux pair. Let m,n be the set of Legendrian curves that are the conormal of a plane curve with a Puiseux pair (m, n), where g.c.d.(m, n) = 1 and m > 2n, with semigroup as generic as possible. We show that the quotient of m,n by the group of contact transformations is a Zariski open set of a weighted projective space. The main tool used in the proof of this theorem is a classification/construction theorem for contact transformation that has since proved useful in other instances. In the second chapter we calculate the limits of tangents of a quasi-ordinary hypersurface. In particular, we show that the set of limits of tangents is, in general, a topological invariant of the hypersurface. In the third chapter we prove a desingularization theorem for Legendrian hypersurfaces that are the conormal of a quasi-ordinary hypersurface. One of the main ingredients of the proof is the calculation of the limits of tangents achieved in chapter two.