Abstract
Given six points on a conic, Pascal’s theorem gives rise to a configuration called the hexagrammum mysticum. It contains 20 Steiner points and 20 Cayley–Salmon lines. By a classical theorem due to von Staudt, the Steiner points fall into 10 conjugate pairs with reference to the conic; but this is not true of the C-S lines for a general choice of six points. We show that the C-S lines are pairwise conjugate precisely when the original sextuple is tri-symmetric. The variety of tri-symmetric sextuples turns out to be arithmetically Cohen–Macaulay of codimension two. We determine its SL2-equivariant minimal resolution.
Acknowledgements
I thank the referee for a thorough reading of the manuscript and several constructive suggestions.
References
[1] P. Aluffi, C. Faber, Linear orbits of d-tuples of points in ℙ1. J. Reine Angew. Math. 445 (1993), 205–220. MR1244973 Zbl 0781.1403610.1515/crll.1993.445.205Search in Google Scholar
[2] H. F. Baker, Principles of geometry. Volume 2. Plane geometry. Cambridge Univ. Press 2010. MR2857757 Zbl 1206.14003Search in Google Scholar
[3] O. Bolza, On binary sextics with linear transformations into themselves. Amer. J. Math. 10 (1887), 47–70. MR1505464 JFM 19.0488.01 JFM 19.0119.0410.2307/2369402Search in Google Scholar
[4] J. Chipalkatti, On Hermite’s invariant for binary quintics. J. Algebra317 (2007), 324–353. MR2360152 Zbl 1130.1403310.1016/j.jalgebra.2007.06.021Search in Google Scholar
[5] J. Chipalkatti, On the coincidences of Pascal lines. Forum Geom. 16 (2016), 1–21. MR3474532 Zbl 1335.51029Search in Google Scholar
[6] J. Conway, A. Ryba, The Pascal mysticum demystified. Math. Intelligencer34 (2012), 4–8. MR2973516 Zbl 1262.5100310.1007/s00283-012-9301-4Search in Google Scholar
[7] H. S. M. Coxeter, Twelve points in PG(5, 3) with 95040 self-transformations. Proc. Roy. Soc. London. Ser. A247 (1958), 279–293. MR0120289 Zbl 0082.3620710.1098/rspa.1958.0184Search in Google Scholar
[8] H. S. M. Coxeter, Projective geometry. Springer 1994. MR1335229 Zbl 0629.51001Search in Google Scholar
[9] I. Dolgachev, Lectures on invariant theory. Cambridge Univ. Press 2003. MR2004511 Zbl 1023.1300610.1017/CBO9780511615436Search in Google Scholar
[10] I. Dolgachev, D. Ortland, Point sets in projective spaces and theta functions. Astérisque no. 165 (1988), 210 pp. (1989). MR1007155 Zbl 0685.14029Search in Google Scholar
[11] J. H. Grace, A. Young, The algebra of invariants. Cambridge Univ. Press 2010. MR2850282 Zbl 1206.1300310.1017/CBO9780511708534Search in Google Scholar
[12] P. Griffiths, J. Harris, Principles of algebraic geometry. Wiley-Interscience 1994. MR1288523 Zbl 0836.1400110.1002/9781118032527Search in Google Scholar
[13] B. Grünbaum, Configurations of points and lines, volume 103 of Graduate Studies in Mathematics. Amer. Math. Soc. 2009. MR2510707 Zbl 1205.5100310.1090/gsm/103Search in Google Scholar
[14] O. Hesse, Ueber die Reciprocität der Pascal–Steinerschen und der Kirkman–Cayley–Salmonschen Sätze von dem Hexagrammum mysticum. J. Reine Angew. Math. 68 (1868), 193–207. MR1579392 Zbl 068.1769cj10.1515/crll.1868.68.193Search in Google Scholar
[15] S. Mukai, H. Umemura, Minimal rational threefolds. In: Algebraic geometry (Tokyo/Kyoto, 1982), volume 1016 of Lecture Notes in Math., 490–518, Springer 1983. MR726439 Zbl 0526.1400610.1007/BFb0099976Search in Google Scholar
[16] D. Pedoe, Geometry. Dover Publications, Inc., New York 1988. MR1017034 Zbl 0716.51002Search in Google Scholar
[17] H. W. Richmond, The figure formed from six points in space of four dimensions. Math. Ann. 53 (1900), 161–176. MR1511085 JFM 31.0547.0210.1007/BF01456032Search in Google Scholar
[18] G. Salmon, A Treatise on Conic Sections. Reprint of the 6th ed. by Chelsea Publishing Co., New York, 2005. Zbl 0211.24002Search in Google Scholar
[19] A. Seidenberg, Lectures in projective geometry. Van Nostrand Company, London 1962. Zbl 0121.37705Search in Google Scholar
[20] J. G. Semple, G. T. Kneebone, Algebraic projective geometry. Oxford Univ. Press 1952. MR0049579 Zbl 0046.38103Search in Google Scholar
[21] B. Sturmfels, Algorithms in invariant theory. Springer 1993. MR1255980 Zbl 0802.1300210.1007/978-3-7091-4368-1Search in Google Scholar
[22] J. J. Sylvester, Note on the historical origin of the unsymmetrical six-valued function of six letters. Philosophical Magazine21 (1861), 369–377. Collected Mathematical Papers, vol. II, p. 264 (see also vol. I, p. 92), Cambridge Univ. Press 1908.10.1080/14786446108643072Search in Google Scholar
[23] C. von Staudt, Ueber die Steinerschen Gegenpunkte, welche durch zwei in eine Curve zweiter Ordnung beschriebene Dreiecke bestimmt sind. J. Reine Angew. Math. 62 (1863), 142–150. Zbl 062.1615cj10.1515/crll.1863.62.142Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- The non-solvable doubly transitive dimensional dual hyperovals
- On the real differential of a slice regular function
- Generalized null 2-type immersions in Euclidean space
- A dual rigidity of the sphere and the hyperbolic plane
- The conjugacy locus of Cayley–Salmon lines
- Steiner’s Porism in finite Miquelian Möbius planes
- Enumeration of complex and real surfaces via tropical geometry
- On complete Yamabe solitons
- Polytopal approximation of elongated convex bodies
- A note on commutative semifield planes
- Quartic surfaces with icosahedral symmetry
Articles in the same Issue
- Frontmatter
- The non-solvable doubly transitive dimensional dual hyperovals
- On the real differential of a slice regular function
- Generalized null 2-type immersions in Euclidean space
- A dual rigidity of the sphere and the hyperbolic plane
- The conjugacy locus of Cayley–Salmon lines
- Steiner’s Porism in finite Miquelian Möbius planes
- Enumeration of complex and real surfaces via tropical geometry
- On complete Yamabe solitons
- Polytopal approximation of elongated convex bodies
- A note on commutative semifield planes
- Quartic surfaces with icosahedral symmetry