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Quadratic embeddings

The quadratic Veronese embedding $ρ$ 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 $M^ρ$. In other words, $ρ$ transforms the closure from quadratic into inear. In this paper we use this property to define "quadratic embeddings". We shall prove that if $ν$ is a quadratic embedding of $PG{n,F)$ into $PG(n',F')$ ($F$ a commutative field), then $ρ^{-1}ν$ 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.