# ALGEBRAIC COMBINATORICS

Twisted quadrics and $\alpha$-flocks
Algebraic Combinatorics, Volume 5 (2022) no. 5, pp. 803-826.

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

The Thas–Walker constructions of conical flocks and flocks of hyperbolic quadrics are extended to their $\alpha$-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 $\alpha$-conical flocks and $\alpha$-twisted hyperbolic flocks. $j$-planes are used to construct two infinite classes of finite $\alpha$-hyperbolic flocks.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.216
Classification: 51E20,  51E14
Keywords: twisted hyperbolic flocks, Klein quadric, j-planes, quasifibrations, T-copies, quaternion division rings
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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