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Additional resources for Algerbaic Geometry and Its Applications: Dedicated to Gilles Lachaud on His 60th Birthday (Series on Number Theory and Its Applications)
But since the vertex figure of t2a5 is [a 2 ,a 2 ], which has only nine vertices, we should expect one of these ten to be special. Such is easily seen to be the case, the special plane being, of course, 123. There is no plane that will make an even triad with every pair of 123, 049, 059, 069, 079, 089. But there are four possibilities if we replace 123 by any other symbol formed with three of the digits 0, 1, 2, 3, 9. , with 039 we can have any of 012, 019, 029, 129. * Noether, Math. Annalen, 14 (1879), 270.
The substitution 0123/456789, for example, interchanges 014 and 234, 456 and 789, but leaves 012 and 345 unaltered. Thus F 123 , when regarded as permuting the joins of opposite vertices ("diameters", say) instead of the vertices themselves, is identical with 0123/456789; and similarly for any Viik. But the interchanges involved when the digit 0 occurs after the stroke in Bath's symbol show that a substitution such as 1234/567890 is not a symmetry of (PA)S. In fact, the diameters 123 and 345 are perpendicular, whereas their transforms 123 and 125 are inclined at ^-TT.
4. 3. 4. 4. 1. 2. 1. 9. 3. 5. 2. 4. 3. 5. 3. 8. Young, L. 2.
Algerbaic Geometry and Its Applications: Dedicated to Gilles Lachaud on His 60th Birthday (Series on Number Theory and Its Applications) by Jean Chaumine, James Hirschfeld, Robert Rolland