Let $GQ(2,4)$ denote the abstract configuration (=incidence structure) consisting of $27$ points and $45$ lines, with $3$ points on leach line and $5$ lines through each point, which can be described in any one of the following ways:

  1. it is the classical generalized quadrangle with parameters $(2,4)$, i.e., it consists of the points and lines lying on a smooth quadric $Q \subseteq \mathbb{P}^5(\mathbb{F}_2)$ defined by a quadratic form of "minus type" (=Witt index $2$; or for definiteness, $x_0^2 + x_0 x_1 + x_1^2 + x_2 x_3 + x_4 x_5 = 0$),

  2. it is the incidence structure obtained by considering the $27$ lines lying on a smooth cubic surface in $\mathbb{P}^3$ (over an algebraically closed field) and intersecting with a plane in general position (the $45$ lines come from intersecting the tritangent planes of the surface).

[See also this related question (note 1) for a combinatorial description of the same incidence structure.]

By an "(extended) Euclidean realization" of $GQ(2,4)$ I mean a set of $27$ points and $45$ lines in the Euclidean plane (resp. the projective completion of it) such that the incidences are precisely those prescribed by $GQ(2,4)$.

Description nº2 above shows that $GQ(2,4)$ indeed has a Euclidean realization, and in fact one in which all $27$ points lie on the same smooth cubic (namely the intersection of the cubic surface):

Question A: Is it the case for every (extended) Euclidean realization of $GQ(2,4)$ that all $27$ points lie on some cubic? (Stronger variant: does it come from the construction described above?)

Question B: Is does there exist an (extended) Euclidean realization of $GQ(2,4)$ having a Euclidean symmetry group of order $\geq 3$?

(The two questions are related in that I have a sort-of-argument why a positive answer to question A implies a negative one to question B.)

More generally, any comment as to Euclidean realizations of $GQ(2,4)$, or pointers to where they have been considered in the literature, are welcome.

  • $\begingroup$ I don't see a 2d realization as it looks like at most 12 of the 45 lines are not parallel to any given line. Perhaps there is a 3d realization involving skew lines? Gerhard "Finite Geometries Fit Me Better" Paseman, 2017.02.18. $\endgroup$ Feb 18, 2017 at 22:00


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.