Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Dimension Formula for the Nucleus of a Veronese Variety

Johannes Gmainer, Hans Havlicek

Abstract

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 $\a\in\V$.

A Dimension Formula for the Nucleus of a Veronese Variety

Abstract

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 -th symmetric power of a finite-dimensional vector space which is spanned by the powers with .

Paper Structure

This paper contains 3 sections, 4 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Veronese Mappings
  3. Nuclei

Key Result

Lemma 1

The vector space ${\bf Y}$ is spanned by $g({\bf X})$ if, and only if, $\#F\geq t$ or $m=\dim {\bf X}-1=0$.

Theorems & Definitions (5)

  • Lemma 1
  • Definition 1
  • Theorem 1
  • Theorem 2
  • Corollary 1