Sub-weakly embedded singular and degenerate polar spaces (Q1360279)

The authors complete the classification of all sub-weak embeddings of orthogonal, symplectic and unitary polar spaces \(\Gamma\) of non-singular rank \(\geq 3\) defined over a commutative field \({\mathbf F}\), where \({\mathbf F}\) is assumed to be perfect if \(\text{char } {\mathbf F} =2\), the polar space is symplectic and the degree of the embedding is \(2\). This is done by showing that every sub-weak embedding of \(\Gamma\) in some projective space \(\text{PG}(d,{\mathbf K})\), where \({\mathbf K}\) is a commutative field, is the projection of a full embedding in some subspace \(\text{PG}(d',{\mathbf K}')\) of \(\text{PG}(d',{\mathbf K})\), where \({\mathbf K'}\) is a subfield of \({\mathbf K}\) and \(\text{PG}(d',{\mathbf K})\) contains \(\text{PG}(d,{\mathbf K})\) as a subspace. A sub-weak embedding of \(\Gamma\) into \(\text{PG}(d,{\mathbf K})\) is a monomorphism \(\theta\) of \(\Gamma\) into the geometry of points and lines of \(\text{PG}(d,{\mathbf K})\) such that (1) the image \(S^\theta\) of the point set \(S\) of \(\Gamma\) generates \(\text{PG}(d,{\mathbf K})\) and (2) for any \(x \in S\), the subspace generated by \(X = \{y^\theta\mid y \in S\) is collinear with \(x\}\) meets \(S^\theta\) exactly in \(X\). A sub-weak embedding is called a full embedding, if for every line \(L\) of \(\Gamma\) all points of \(\text{PG}(d,{\mathbf K})\) on the line \(L^\theta\) have an inverse image under \(\theta\).