×

A short note on Cayley-Salmon equations. (English) Zbl 1453.14130

In the paper under review, the authors study a Cayley-Salmon equation for a smooth cubic surface \(S \subset \mathbb{P}^{3}\) which is an expression of the form \(\ell_{1}\ell_{2}\ell_{3} - m_{1}m_{2}m_{3} = 0\) such that the zero set is \(S\) and \(\ell_{i}\) and \(m_{j}\) are homogeneous linear forms in four variables. This expression was firstly used by Cayley and Salmon in order to study the incidence relations of the \(27\) lines on \(S\). First of all, the authors reproduce proofs of the fact that \(S\) contains \(27\) lines, \(45\) tritangent planes that intersect \(S\) at \(3\) distinct lines, and \(120\) pairs of triples of tritangent planes where each triple of planes altogether intersects \(S\) in nine lines. The conclusion of the classical constructions is that knowing the equations for the \(27\) lines one can compute the Cayley-Salmon equations, and conversely, from a Cayley-Salmon equation it is possible to derive equations for the \(27\) lines. It is well-known that there are \(120\) essentially distinct Cayley-Salmon equations for \(S\). In the second part of the paper the authors illustrate the explicit calculation to obtain these equations and they apply it to the Clebsch surface and to the octanomial model. Moreover, they show that these \(120\) Cayley-Salmon equations can be directly computed using tropical geometry methods developed recently by M. A. Cueto and A. Deopurkar [“Anticanonical tropical cubic del Pezzos contain exactly 27 lines”, Preprint, arXiv:1906.08196].

MSC:

14N05 Projective techniques in algebraic geometry
14J26 Rational and ruled surfaces

Software:

Macaulay2
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] John Baez,27 Lines on a Cubic Surface,https://blogs.ams.org/ visualinsight/2016/02/15/27-lines-on-a-cubic-surface/, 2016, Accessed: 2019-11-22.
[2] Mauro C Beltrametti - Ettore Carletti - Dionisio Gallarati - Giacomo Monti Bragadin,Lectures on Curves, Surfaces and Projective Varieties, (2009). · Zbl 1180.14001
[3] C. C. Bramble,A Collineation Group Isomorphic with the Group of the Double Tangents of the Plane Quartic, Amer. J. Math. 40 no. 4 (1918), 351-365. · JFM 46.0891.01
[4] Anita Buckley - Tomaˇz Koˇsir,Determinantal representations of smooth cubic surfaces, Geom. Dedicata 125 (2007), 115-140. · Zbl 1117.14038
[5] Arthur Cayley,On the Triple Tangent Planes of Surfaces of the Third Order, The Cambridge and Dublin mathematical journal 4 (1849), 118-132.
[6] Maria Angelica Cueto - Anand Deopurkar,Anticanonical tropical cubic del Pezzos contain exactly 27 lines, arXiv 1906.08196 (2019).
[7] Igor V Dolgachev,Classical Algebraic Geometry: A Modern View, Cambridge University Press, 2012. · Zbl 1252.14001
[8] Gerd Fischer,Plane Algebraic Curves, vol. 15, American Mathematical Soc., 2001. · Zbl 0971.14026
[9] Daniel R. Grayson - Michael E. Stillman,Macaulay2, a software system for research in algebraic geometry, Available athttp://www.math.uiuc.edu/ Macaulay2/.
[10] Robin Hartshorne,Algebraic geometry, vol. 52, Springer Science & Business Media, 2013. · Zbl 0367.14001
[11] Archibald Henderson,The Twenty-Seven Lines upon the Cubic Surface, Cambridge University Press, 1911. · JFM 42.0661.01
[12] Marta Panizzut - Emre Can Sert¨oz - Bernd Sturmfels,An Octanomial Model for Cubic Surfaces, arXiv preprint arXiv:1908.06106 (2019).
[13] Miles Reid,Undergraduate Algebraic Geometry, Cambridge University Press Cambridge, 1988. · Zbl 0701.14001
[14] Qingchun Ren - Steven V. Sam - Gus Schrader - Bernd Sturmfels,The Universal Kummer Threefold, Experimental Mathematics 22 no. 3 (2013), 327-362 (en). · Zbl 1312.14103
[15] G. Salmon,A treatise on the analytical geometry of three dimensions. Revised by R. A. P. Rogers. Fifth edition. In two volumes. Vol. I., London: Longmans, Green and Co.; Dublin: Hodges, Figgis and Co. XXII u. 470 S. (1912)., 1912. · JFM 43.0646.06
[16] G. Salmon,A treatise on the analytic geometry of three dimensions.5thedition. Edited by R. A. P. Rogers. Vol. II., London: Longmans, Green and Co.(1915)., 1915. · JFM 45.0807.05
[17] W.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.