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Groups Associated with Certain Loci In [5]

  • A. F. Horadam (a1)

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If x- (i = 1, 2, … , 6) are homogeneous co-ordinates in [5] (the complex projective space of five dimensions), then the equation

1.1

represents the well-known (11) Perazzo cubic primal P4 3 of order 3 and dimension 4. With it is associated the Segre cubic threefold S3 3 (12); specifically, S3 3 is the section of P4 3 by a tangent [4].

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References

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1. Baker, H. W., Segre's ten nodal cubic primal in space of four dimensions and del Pezzo's surface in five dimensions, J. Lond. Math. Soc, 6 (1931), 176185.
2. Baker, H. W., Principles of Geometry, vol. 4 (Cambridge, 1950), pp. 148, 151.
3. Bolt, Beverley, Room, T. G., and Wall, G. E., On the Clifford collineation, transform and similarity groups I, II, J. Aust. Math. Soc, 2 (1961), 6096.
4. Bolt, Beverley, On the Clifford collineation, transform and similarity groups III. Generators and involutions, J. Aust. Math. Soc, 2 (1962), 334344.
5. Clebsch, A., Leçons sur la géométrie (Paris, 1880), pp. 232233.
6. Horadam, A. F., A locus in [8] invariant under a group of order 51840 × 81, Quart. J. Math. (Oxford), (2) 8 (1957), 241259.
7. Horadam, A. F., Projection of an invariant locus in [8] from a solid lying on it, Quart. J. Math. (Oxford), (2) 9 (1958), 8187.
8. Horadam, A. F., Involutions associated with the Burkhardt configuration in [4], Can. J. Math. 11 (1959), 1833.
9. Horadam, A. F., Clifford groups in the plane, Quart. J. Math. (Oxford), (2) 10 (1959), 294295.
10. Horadam, A. F., Clifford matrices and the Hessian group, Rend, del Circ Mat. di Palermo (2), 10 (1961), 347352.
11. Perazzo, U., Sopra una forma con 9 rette doppie dello spazio a cinque dimensioni, Atti Reale Ace Sci. Torino, 86 (1900-1), 891916.
12. Segre, C., Sidle varieta cubiche dello spazio a quattro dimensioni e su certi sistemi di rette e certe superficie dello spazio ordinario, Mem. Reale Ace della Scienze Torino, (2) 39 (1889), 348.
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