Constructing non-regular algebraic spreads with asymplecticly complemented regulization

Rolf Riesinger

Abstract: We give an application of the second extension of the Thas-Walker construction and exhibit a $4$-parameter family $\cal F$ of explicit examples of spreads of $\mbox{{\rm PG}}(3,\Bbb R)$ with asymplecticly complemented regulization. In $\cal F$ there are symplectic spreads and also asymplectic algebraic spreads. A spread $\cal S$ of $\mbox{{\rm PG}}(3,\Bbb R)$ is called rigid if, apart from the identity, there exists no collineation leaving $\cal S$ invariant; a rigid spread $\cal S$ is said to be hyperrigid if there exists no duality leaving $\cal S$ invariant. The family $\cal F$ contains hyperrigid algebraic spreads as well as rigid algebraic spreads which are not hyperrigid.

Keywords: spread, algebraic spread, hyperrigid spread, $4$-dimensional translation plane

Classification (MSC2000): 51A40, 51H10, 51M30

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