ISSN:
1572-9168
Keywords:
polar space
;
embedding.
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We show that every sub-weak embedding of any singular (degenerate or not) orthogonal or unitary polar space of non-singular rank at least 3 in a projective space PG $$(d,\mathbb{K}) $$ , $$\mathbb{K} $$ a commutative field, is the projection of a full embedding in some subspace PG $$\left( \overline {d}, \mathbb{F} \right) $$ of PG $$\left( \overline {d}, \mathbb{K} \right) $$ , where PG $$\left( \overline {d}, \mathbb{K} \right) $$ contains PG $$(d,\mathbb{K}) $$ and $$\mathbb{F} $$ is a subfield of $$\mathbb{K} $$ . The same result is proved in the symplectic case under the assumption that the field over which the polarity is defined is perfect if the characteristic is 2 and if each secant line of the embedded polar space Γ contains exactly two points of Γ. This completes the classification of all sub-weak embeddings of orthogonal, symplectic and unitary polar spaces (singular or not; degenerate or not) of non-singular rank at least 3 and defined over a commutative field $$\mathbb{F}\prime $$ , where in the characteristic 2 case $$\mathbb{F}\prime $$ is perfect if the polar space is symplectic and the degree of the embedding is 2.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1017997605039
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