Quadratic Embeddings

Abstract: The quadratic Veronese embedding $\rho$ maps the point set $\cal P$ 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 $\cal M\subset\cal P$, then the intersection of all quadrics containing $\cal M$ is the inverse image of the linear closure of ${\cal M}^{\rho}$. In other words, $\rho$ transforms the closure from quadratic into linear. 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.

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