Bitangents of a quartic

The Trott curve, another curve with 28 real bitangents, is the set of points (x,y) satisfying the degree four polynomial equation

${\displaystyle \displaystyle 144(x^{4}+y^{4})-225(x^{2}+y^{2})+350x^{2}y^{2}+81=0.}$

These points form a nonsingular quartic curve that has genus three and that has twenty-eight real bitangents.[3]

Like the examples of Plücker and of Blum and Guinand, the Trott curve has four separated ovals, the maximum number for a curve of degree four, and hence is an M-curve. The four ovals can be grouped into six different pairs of ovals; for each pair of ovals there are four bitangents touching both ovals in the pair, two that separate the two ovals, and two that do not. Additionally, each oval bounds a nonconvex region of the plane and has one bitangent spanning the nonconvex portion of its boundary.

The dual curve to a quartic curve has 28 real ordinary double points, dual to the 28 bitangents of the primal curve.

The 28 bitangents of a quartic may also be placed in correspondence with symbols of the form

${\displaystyle {\begin{bmatrix}a&b&c\\d&e&f\\\end{bmatrix}}}$

where a, b, c, d, e, f are all zero or one and where

${\displaystyle ad+be+cf=1\ (\operatorname {mod} \ 2).}$[4]

There are 64 choices for a, b, c, d, e, f, but only 28 of these choices produce an odd sum. One may also interpret a, b, c as the homogeneous coordinates of a point of the Fano plane and d, e, f as the coordinates of a line in the same finite projective plane; the condition that the sum is odd is equivalent to requiring that the point and the line do not touch each other, and there are 28 different pairs of a point and a line that do not touch.

The points and lines of the Fano plane that are disjoint from a non-incident point-line pair form a triangle, and the bitangents of a quartic have been considered as being in correspondence with the 28 triangles of the Fano plane.[5] The Levi graph of the Fano plane is the Heawood graph, in which the triangles of the Fano plane are represented by 6-cycles. The 28 6-cycles of the Heawood graph in turn correspond to the 28 vertices of the Coxeter graph.[6]

The 28 bitangents of a quartic also correspond to pairs of the 56 lines on a degree-2 del Pezzo surface,[5] and to the 28 odd theta characteristics.

The 27 lines on the cubic and the 28 bitangents on a quartic, together with the 120 tritangent planes of a canonic sextic curve of genus 4, form a "trinity" in the sense of Vladimir Arnold, specifically a form of McKay correspondence,[7][8][9] and can be related to many further objects, including E₇ and E₈, as discussed at trinities.

• Blum, R.; Guinand, A. P. (1964). "A quartic with 28 real bitangents". Canadian Mathematical Bulletin. 7 (3): 399–404. doi:10.4153/cmb-1964-038-6.
• Cayley, Arthur (1879), "On the bitangents of a quartic", Salmon's Higher Plane Curves, pp. 387–389. In The collected mathematical papers of Arthur Cayley, Andrew Russell Forsyth, ed., The University Press, 1896, vol. 11, pp. 221–223.
• Gray, Jeremy (1982), "From the history of a simple group", The Mathematical Intelligencer, 4 (2): 59–67, doi:10.1007/BF03023483, MR 0672918, S2CID 14602496. Reprinted in Levy, Silvio, ed. (1999), The Eightfold Way, MSRI Publications, vol. 35, Cambridge University Press, pp. 115–131, ISBN 0-521-66066-1, MR 1722415.
• Manivel, L. (2006), "Configurations of lines and models of Lie algebras", Journal of Algebra, 304 (1): 457–486, arXiv:math/0507118, doi:10.1016/j.jalgebra.2006.04.029, S2CID 17374533.
• McKay, John; Sebbar, Abdellah (2007). "Replicable Functions: An Introduction". Frontiers in Number Theory, Physics, and Geometry II. pp. 373–386. doi:10.1007/978-3-540-30308-4_10. ISBN 978-3-540-30307-7.
• Plücker, J. (1839), Theorie der algebraischen Curven: gegrundet auf eine neue Behandlungsweise der analytischen Geometrie, Berlin: Adolph Marcus.
• Riemann, G. F. B. (1876), "Zur Theorie der Abel'schen Funktionen für den Fall p = 3", Ges. Werke, Leipzig, pp. 456–472{{citation}}: CS1 maint: location missing publisher (link). As cited by Cayley.
• Shioda, Tetsuji (1995), "Weierstrass transformations and cubic surfaces" (PDF), Commentarii Mathematici Universitatis Sancti Pauli, 44 (1): 109–128, MR 1336422
• Trott, Michael (1997), "Applying GroebnerBasis to Three Problems in Geometry", Mathematica in Education and Research, 6 (1): 15–28.