Quadratic embeddings
نویسندگان
چکیده
The quadratic Veronese embedding ρ maps the point set P of PG(n, F ) into the point set of PG( (n+2 2 ) − 1, F ) (F a commutative field) and has the following well-known property: If M ⊂ P, then the intersection of all quadrics containing M is the inverse image of the linear closure of Mρ. In other words, ρ transforms the closure from quadratic into linear. 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−1ν 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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