Rationale Normkurven in Räumen mit positiver Charakteristik

Abstract

Let PG(n,F) be the n-dimensional projective space on F "("n "+ "1), where n is not less than two and F is a (commutative) field. Each normal rational curve is projectively equivalent to the union of the set {F(1, t,... t "n} (t an element of F) an the point F(0, 0, ..., 1). If the characteristic of the ground field F equals zero, we can define the osculating subspaces of the curve by the help of formal derivation. In the case of non-zero characteristic this is not possible and therefore we introduce a non-iterative derivation of polynomials. The k-nucleus of a normal rational curve is defined as intersection over all the osculating k-subspaces of the curve. It is well known that for characteristic zero all nuclei are empty, whereas characteristic p = 0 leads in most cases to non-trivial nuclei. The geometric properties of a k-nucleus are closely related with binomial coefficients that vanish modulo p and, on the other hand, with the representations of the integers n, n+1, and k in base p. The zero entries of Pascal's triangle modulo p fall into various classes and the corresponding partition gives rise to three functions which are described in the first chapter. With the help of these functions we are able to give a formula expressing the dimensions of all the nuclei. Normal rational curves admit a group G of collineations preserving all osculating subspaces. We describe the set of all G-invariant subspaces and show that this set does not form a chain - in contrast to the subset of all the k-nuclei.