Twisted quadrics and $\alpha $-flocks

Johnson, Norman L.

doi:10.5802/alco.216

Twisted quadrics and

α

-flocks

Johnson, Norman L.¹

¹ University of Iowa 750 E. Foster Rd. #306 Iowa City IA 52245

Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 803-826.

Abstract

In this article, we provide a general study of what we call twisted quadrics and consider flocks of the variant of $α$ -conics and $α$ -hyperbolic quadrics. We extend the notion of the Klein quadric to what we call an $α$ -Klein quadric. Blended kernel translation planes are defined and analysed when considering $α$ -conical flocks and $α$ -twisted hyperbolic flocks.

The Thas–Walker constructions of conical flocks and flocks of hyperbolic quadrics are extended to their $α$ -analogues. Using the idea that any derivable net can be embedded into a $3$ -dimensional projective space over a skewfield, allows us to formulate what might be called a projective version of work previously given in an algebraic framework. The theory of deficiency one flocks is extended to both $α$ -conical flocks and $α$ -twisted hyperbolic flocks. $j$ -planes are used to construct two infinite classes of finite $α$ -hyperbolic flocks.

Received: 2021-01-28
Revised: 2021-12-17
Accepted: 2021-12-21
Published online: 2022-11-09

DOI: 10.5802/alco.216

Classification: 51E20, 51E14
Keywords: twisted hyperbolic flocks, Klein quadric, j-planes, quasifibrations, T-copies, quaternion division rings

Author's affiliations:

Johnson, Norman L. ¹

¹ University of Iowa 750 E. Foster Rd. #306 Iowa City IA 52245

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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Johnson, Norman L. Twisted quadrics and $\alpha $-flocks. Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 803-826. doi : 10.5802/alco.216. https://alco.centre-mersenne.org/articles/10.5802/alco.216/

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