It is known that every lax projective embedding e : Gamma -> PG(V) of a point-line geometry Gamma admits a hull, namely a projective embedding (e) over tilde: Gamma -> PG((V) over tilde) uniquely determined up to isomorphisms by the following property: V and (V) over tilde are defined over the same skewfield, say K, there is morphism of embeddings (f) over tilde : (e) over tilde -> e and, for every embedding e' : Gamma -> PG(V') with V' defined over K, if there is a morphism g : e' - e then a morphism f : e' also exists such that (f) over tilde = gf. If e = (e) over tilde then we say that e is dominant. Clearly, hulls are dominant. Let now Gamma be a non-degenerate polar space of rank n >= 3. We shall prove the following: A lax embedding e: Gamma -> PG(V) is dominant if and only if, for every geometric hyperplane H of Gamma, e(H) spans a hyperplane of PG(V). We shall also give some applications of the above result.

Pasini, A. (2005). Dominant lax embeddings of polar spaces. BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY SIMON STEVIN, 12(5), 871-882 [10.36045/bbms/1136902622].

Dominant lax embeddings of polar spaces

PASINI, ANTONIO
2005

Abstract

It is known that every lax projective embedding e : Gamma -> PG(V) of a point-line geometry Gamma admits a hull, namely a projective embedding (e) over tilde: Gamma -> PG((V) over tilde) uniquely determined up to isomorphisms by the following property: V and (V) over tilde are defined over the same skewfield, say K, there is morphism of embeddings (f) over tilde : (e) over tilde -> e and, for every embedding e' : Gamma -> PG(V') with V' defined over K, if there is a morphism g : e' - e then a morphism f : e' also exists such that (f) over tilde = gf. If e = (e) over tilde then we say that e is dominant. Clearly, hulls are dominant. Let now Gamma be a non-degenerate polar space of rank n >= 3. We shall prove the following: A lax embedding e: Gamma -> PG(V) is dominant if and only if, for every geometric hyperplane H of Gamma, e(H) spans a hyperplane of PG(V). We shall also give some applications of the above result.
Pasini, A. (2005). Dominant lax embeddings of polar spaces. BULLETIN OF THE BELGIAN MATHEMATICAL SOCIETY SIMON STEVIN, 12(5), 871-882 [10.36045/bbms/1136902622].
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11365/7315
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