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Some loci in S5 associated with systems of conics in a plane

Published online by Cambridge University Press:  14 November 2011

A. J. Knight
A. J. Knight
Affiliation:
Mathematics Division, University of Sussex
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Synopsis

If the conic α, a0x2 + a1y2 + a2z2 + 2a3yz + 2a4zx + 2a5xy=0, is represented by the point (a0, a1, a2, a3, a4, a5) of S5, the transforms of α by projectivities that fix a second conic ω will generally be represented by points of a threefold in S5 containing (a0, a1,…, a5). It is shown that this threefold is in general a rational sextic belonging to an ∞2 family, that is composed of ∞1 sets of projectively equivalent threefolds. Special, exceptional members of the family are discussed.

Equations for the threefold are found in terms of the mutual projection invariants of ω and α.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 84 , Issue 1-2 , 1979 , pp. 93 - 107
Copyright
Copyright © Royal Society of Edinburgh 1979

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References

1Semple, J. G. and Roth, L.. Introduction to Algebraic Geometry (Oxford: Clarendon, 1949).Google Scholar
2Todd, J. A.. Projective and Analytical Geometry (New York: Pitman, 1946).Google Scholar
3Knight, A. J.. Conies related by a pentagram: a problem of J. E. Reeve. Proc. Roy. Soc. Edinburgh Sect. A 84, (1979), 109–115.CrossRefGoogle Scholar