Quadratic embeddings

TL;DR

This paper explores quadratic embeddings in projective geometry, demonstrating how the quadratic Veronese embedding transforms quadratic closure into linear closure and characterizing these embeddings geometrically.

Contribution

It introduces the concept of quadratic embeddings, proves their properties, and provides a geometric characterization of the quadratic Veronese embedding.

Findings

Quadratic Veronese embedding transforms quadratic closure into linear closure.

Quadratic embeddings induce an injective homomorphism between fields in most cases.

A geometric characterization of the quadratic Veronese embedding is established.

Abstract

The quadratic Veronese embedding $\rho$ maps the point set $P$ of $\PG{n,F)$ into the point set of $PG({n+2 \choose 2}-1, F$ ($F$ a commutative field) and has the following well-known property: If $M\subset P$, then the intersection of all quadrics containing $M$ is the inverse image of the linear closure of $M^{\rho}$. In other words, $\rho$ transforms the closure from quadratic into inear. In this paper we use this property to define "quadratic embeddings". We shall prove that if $\nu$ is a quadratic embedding of $PG{n,F)$ into $PG(n',F')$ ($F$ a commutative field), then $\rho^{-1}\nu$ is dimension-preserving. Moreover, up to some exceptional cases, there is an injective homomorphism of $F$ into $F'$. An additional regularity property for quadratic embeddings allows us to give a geometric characterization of the quadratic Veronese embedding.