Veronese varieties over fields with non-zero characteristic: a survey

Non-zero characteristic of the (commutative) ground field F heavily influences the geometric properties of Veronese varieties and, in particular, normal rational curves. Best known is probably the fact that, in case of characteristic two, all tangents of a conic are concurrent. This has lead to the concept of a nucleus. However, it seems that there are essentially distinct definitions. Some authors, like J.A. Thas [38], use this term to denote a point which extends a normal rational curve to an (q + 2)-arc (F a finite field of even order q), others, like A. Herzer [23], use the same term for the intersection of all osculating hyperplanes of a Veronese variety. In order to overcome this difference of terminology we introduce the term (r, k)-nucleus. The two types of nuclei mentioned above are just particular examples fitting into this general concept. Each nucleus is an invariant subspace, i.e. a subspace in the ambient space of a Veronese variety which is fixed (as a set of points) under the group of automorphic collineations of the variety. However, an invariant subspace needs not be a nucleus. In the present survey we collect some recent results on nuclei of Veronese varieties and invariant subspaces of normal rational curves. We must assume, however, that the ground field is not “too small”, since otherwise a Veronese variety is like dust: “few points” in some “high-dimensional” space. Nuclei and invariant subspaces do not appear in classical textbooks on Veronese varieties (F = R,C), since for characteristic zero all invariant subspaces are trivial. If the ground field has characteristic p > 0, then geometric properties of invariant subspaces are closely related tomultinomial coefficients that vanish modulo p and to the representations of certain integers in base p. In order to illustrate this connection some results on binomial and multinomial coefficients are gathered in Chapters 2 and 4.