Are minimal degree rational curves determined by their tangent vectors?
arXiv:math/0206193
Abstract
Let X be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this setup, characterization and classification problems lead to the natural question: "Given two points on X, how many minimal degree rational curve are there which contain those points?". A recent answer to this question led to a number of new results in classification theory. As an infinitesimal analogue, we ask "How many minimal degree rational curves exist which contain a prescribed tangent vector?" In this paper, we give sufficient conditions which guarantee that every tangent vector at a general point of X is contained in at most one rational curve minimal degree. As an immediate application, we obtain irreducibility criteria for the space of minimal rational curves.
The main theorems were improved; they now work for families of rational curves that are not necessarily irreducible. For this, it was necessary to change the proofs and the exposition substantially. An optimized PDF file with additional graphics is available at http://btm8x5.mat.uni-bayreuth.de/~kebekus/publications-e.html