A Dimension Formula for the Nucleus of a Veronese Variety

TL;DR

This paper derives an explicit formula for the dimension of the nucleus of a Veronese variety over fields of any characteristic, generalizing previous conditions for its emptiness.

Contribution

It provides a new, explicit dimension formula for the nucleus of a Veronese variety applicable to arbitrary field characteristics.

Findings

Derived a formula for the nucleus dimension in arbitrary characteristic

Connected the nucleus dimension to symmetric powers of vector spaces

Generalized previous conditions for the nucleus to be empty

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 $\a\in\V$.