Classical Generalized Quadrangles
When looking at the different cases for polar spaces of rank at least three, and extrapolating them to rank 2, one finds these (finite) generalized quadrangles :
- A hyperbolic quadric, a parabolic quadric and an elliptic quadric are the only possible quadrics in projective spaces over finite fields with projective index 1. We find these parameters respectively :
- (this is just a grid)
- A hermitian variety has projective index 1 if and only if n is 3 or 4. We find :
- A symplectic polarity in has a maximal isotropic subspace of dimension 1 if and only if . Here, we find a generalized quadrangle, with .
The generalized quadrangle derived from is always isomorphic with the dual of, and they are both self-dual and thus isomorphic to each other if and only if is even.
Read more about this topic: Generalized Quadrangle
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