A k-nucleus of a normal rational curve in PG(n, F ) is the intersection over all k-dimensional osculating subspaces of the curve (k ∈ {−1, 0, . . . , n − 1}). It is well known that for characteristic zero all nuclei are empty. In case of characteristic p > 0 and #F ≥ n the number of non-zero digits in the representation of n + 1 in base p equals the number of distinct nuclei. An explicit formula for the dimensions of k-nuclei is given for #F ≥ k + 1.

The nucleus of a Veronese variety is the intersection of all its osculating hyperplanes. Various authors have given necessary and sufficient conditions for the nucleus to be empty. We present an explicit formula for the dimension of this nucleus for arbitrary characteristic of the ground field. As a corollary, we obtain a dimension formula for that subspace in the t-th symmetric power of a finite-dimensional vector space V which is spanned by the powers a t with a ∈ V.

Each normal rational curve in P G(n, F) admits a group P L( ) of automorphic collineations. It is well known that for characteristic zero only the empty and the entire subspace are P L( )-invariant. In the case of characteristic p > 0 there may be further invariant subspaces. For #F ≥ n+2, we give a construction of all P L( )-invariant subspaces. It turns out that the corresponding lattice is totally ordered in special cases only.

Let F be Cayley's ruled cubic surface in a projective three-space over any commutative field K. We determine all collineations fixing F , as a set, and all cubic forms defining F . For both problems the cases |K| = 2, 3 turn out to be exceptional. On the other hand, if |K| ≥ 4 then the set of simple points of F can be endowed with a nonsymmetric distance function. We describe the corresponding circles, and we establish that each isometry extends to a unique projective collineation of the ambient space.

Abstract. In this paper we study a class of plane self-affine lattice tiles that are defined using polyominoes. In particular, we characterize which of these tiles are homeomorphic to a closed disk. It turns out that their topological structure depends very sensitively on their defining parameters.In order to achieve our results we use an algorithm of Scheicher and the second author which allows to determine neighbors of tiles in a systematic way as well as a criterion of Bandt and Wang, with that we can check disk-likeness of a self-affine tile by analyzing the set of its neighbors.