On symplectic polar spaces over non-perfect fields of characteristic 2

Abstract

Given a field 𝕂 of characteristic 2 and an integer n ≥ 2, let W(2n − 1, 𝕂) be the symplectic polar space defined in PG(2n − 1, 𝕂) by a non-degenerate alternating form of V(2n, 𝕂) and let Q(2n, 𝕂) be the quadric of PG(2n, 𝕂) associated to a non-singular quadratic form of Witt index n. In the literature it is often claimed that W(2n − 1, 𝕂) ≅ Q(2n, 𝕂). This is true when 𝕂 is perfect, but false otherwise. In this article, we modify the previous claim in order to obtain a statement that is correct for any field of characteristic 2. Explicitly, we prove that W(2n − 1, 𝕂) is indeed isomorphic to a non-singular quadric Q, but when 𝕂 is non-perfect the nucleus of Q has vector dimension greater than 1. So, in this case, Q(2n, 𝕂) is a proper subgeometry of W(2n − 1, 𝕂). We show that, in spite of this fact, W(2n − 1, 𝕂) can be embedded in Q(2n, 𝕂) as a subgeometry and that this embedding induces a full embedding of the dual DW(2n − 1, 𝕂) of W(2n − 1, 𝕂) into the dual DQ(2n, 𝕂) of Q(2n, 𝕂).