• N S NARASIMHA SASTRY

Articles written in Proceedings – Mathematical Sciences

• Elliptic ovoids and their rosettes in a classical generalized quadrangle of even order

Let $\mathcal{Q}_0$ be the classical generalized quadrangle of order $q = 2^{n}(n \geq 2)$ arising from a non-degenerate quadratic form in a 5-dimensional vector space defined over a finite field of order $q$. We consider the rank two geometry $\mathcal{X}$ having as points all the elliptic ovoids of $\mathcal{Q}^0$ and as lines the maximal pencils of elliptic ovoids of $\mathcal{Q}_0$ pairwise tangent at the same point. We first prove that $\mathcal{X}$ is isomorphic to a 2-fold quotient of the affine generalized quadrangle $\mathcal{Q} \backslash \mathcal{Q}_0$, where $\mathcal{Q}$ is the classical $(q, q^2)$- generalized quadrangle admitting $\mathcal{Q}_0$ as a hyperplane. Further, we classify the cliques in the collinearity graph $\Gamma$ of $\mathcal{X}$. We prove that any maximal clique in $\Gamma$ is either a line of $\mathcal{X}$ or it consists of 6 or 4 points of $\mathcal{X}$ not contained in any line of $\mathcal{X}$, accordingly as $n$ is odd or even.We count the number of cliques of each type and show that those cliques which are not contained in lines of $\mathcal{X}$ arise as subgeometries of $\mathcal{Q}$ defined over $\mathbb{F}_2$

• Ovoidal fibrations in $PG(3, q)$, $q$ even

Given a partition of a projective 3-space of odd cardinality by a set of ovoids,a line secant to one of the ovoids of the partition, and its polar relative to the symplectic polarity on the projective 3-space defined by this ovoid, are tangent to distinct ovoids of the partition (Theorem 2). The proof uses the fact that the radical of the linear code generated by the duals of the hyperbolic quadrics in a symplectic generalized quadrangle is of codimension one (Theorem 4).

• # Proceedings – Mathematical Sciences

Volume 130, 2020
All articles
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• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019