ORTHOGONAL, SYMPLECTIC AND UNITARY POLAR SPACES SUB-WEAKLY EMBEDDED IN PROJECTIVE-SPACE

We show that every sub-weak embedding of any non-singular orthogonal o r unitary polar space of rank at least 3 in a projective space PG(d, K ), K a commutative field, is a full embedding in some subspace PG(d, F ), where F is a subfield of K; the same theorem is proved for every su b-weak embedding of any non-singular symplectic polar space of rank at least 3 in PG(d, K), where the field F' over which the symplectic pol arity is defined is perfect in the case that the characteristic of F' is two and the secant lines of the embedded polar space Gamma contain exactly two points of Gamma. This generalizes a result announced by LE FEVRE-PERCSY [5] more than ten years ago, but never published. We also show that every quadric defined over a subfield F of K extends unique ly to a quadric over the groundfield K, except in a few well-known cas es.