Anders Lundman: Higher order gauss maps

Consider a projective variety X over the complex numbers. The classical Gauss map is a rational morphism that assigns to a smooth point on X the projective tangent space at that point. A classical theorem says that if X is smooth, then the Gauss map has finite fibers, unless X is projective space. In this talk we will consider a generalisation of the Gauss map by considering osculating spaces, which are linear spaces tangent to higher order. For example the first osculating space at a point p is simply the classical projective tangent space at p. We will concentrate on varieties enjoying the property that the k-th osculating space is of maximal possible dimension at every general point. The Gauss map of order k is then defined as the rational map sending a point in X to the k-th osculating space at that point. Our main result is that, as in the classical case, the Gauss map of order k is finite whenever the k-th osculating space is of maximal dimension at all points of X unless X is the k-th Veronese embedding of projective space. Moreover we will give a combinatorial description of these maps in the case when X is a toric variety.