The image of the Veronese mapping is a determinantal variety; an elementary proof

TL;DR

This paper provides an elementary proof that the image of the Veronese map is a determinantal variety defined by 2-minors of a matrix, clarifying its geometric structure.

Contribution

It offers a simple, elementary proof that the Veronese image is a determinantal variety, enhancing understanding of its algebraic properties.

Findings

Veronese image is a determinantal variety

Defined by 2-minors of a matrix

Elementary proof approach

Abstract

Let $\nu_d : \mathbb{P}^n \longrightarrow \mathbb{P}^N$ be the Veronese mapping of degree $d$ where $N = {n+d \choose n} -1$. By an elementary approach it is shown that $\nu_d$ is an isomorphism of $\mathbb{P}^n$ onto the projective variety defined by the $2$-minors of a matrix.