Nuclei of normal rational curves

Abstract

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.

A dimension formula for the nucleus of a Veronese variety

“…, e m ) ∈ E t m such that the multinomial coefficient t e 0 ,e 1 ,...,em is not divisible by the prime p equals Theorem 2 has been established by H. Timmermann [12, 4.15] for normal rational curves V t 1 . See also [4]. From (5), (15), and Lemma 1 the symmetric powers a * t with a * ∈ X * cannot generate S t X * when #F < t. So here the nucleus of a Veronese variety V t m is non-empty.…”

“…a Veronese image of a projective line. Another proof of that formula and further references can be found in [4]. See also J.A.…”

Veronese Varieties over Fields with non-zero Characteristic: A Survey

“…Definition 1 [12] A pair (n, j) = ( n σ , j σ ) of non-negative integers with j ≤ n, n j ≡ 0 (mod p), and…”

Pascal’s Triangle, Normal Rational Curves, and their Invariant Subspaces

“…In the case of non-zero characteristic all Veronese varieties with empty nucleus have been determined independently by Timmermann [9,10], Herzer [6], and Karzel [8]. In [10] and [4] one can find an explicit formula for the dimension of the nucleus of a normal rational curve; in [3] this is generalized to arbitrary Veronese varieties. The term nucleus can be extended in the following way [4]: define the intersection over all k-dimensional osculating subspaces of the curve to be a k-nucleus.…”

“…In [10] and [4] one can find an explicit formula for the dimension of the nucleus of a normal rational curve; in [3] this is generalized to arbitrary Veronese varieties. The term nucleus can be extended in the following way [4]: define the intersection over all k-dimensional osculating subspaces of the curve to be a k-nucleus. Obviously, these subspaces are further examples of P L( )-invariant subspaces.…”