Twenty-seven questions about the cubic surface. (English) Zbl 1454.14104

Summary: We present a collection of research questions on cubic surfaces in 3-space. These questions inspired the present collection of papers. This article serves as the introduction to the issue. The number of questions is meant to match the number of lines on a cubic surface. We end with a list of problems that are open.


14J26 Rational and ruled surfaces
14Q10 Computational aspects of algebraic surfaces
Full Text: DOI arXiv


[1] H. Abo, A. Seigal and B. Sturmfels:Eigenconfigurations of tensors, Algebraic and Geometric Methods in Discrete Mathematics, Contemporary Mathematics 685, Amer. Math. Soc., Providence, RI (2017) 1-25. · Zbl 1360.15011
[2] D. Allcock, J. A. Carlson, D. Toledo:The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Algebraic Geom.11(2002), 659-724. · Zbl 1080.14532
[3] D. Allcock, J.A. Carlson, D. Toledo:Hyperbolic geometry and moduli of real cubic surfaces. Ann. Sci. ´Ec. Norm. Sup´er. (4) 43 (2010), no. 1, 69-115. · Zbl 1187.14043
[4] H. Baker:Principles of Geometry, vols.1-6., Cambridge University Press, 1922.
[5] N.D. Beklemishev:Invariants of cubic forms of four variables, Vestnik Moscow Univ. Ser. I Mat. Mekh.2(1982) 42-49. · Zbl 0498.14006
[6] M. Bernal, D. Corey, M. Donton-Bury, N. Fujita and G. Merz:Khovanskii bases of Cox-Nagata rings and tropical geometry, Combinatorial Algebraic Geometry, 133-157, Fields Inst. Communications80, Springer, New York. · Zbl 1390.14028
[7] P. Breiding and S. Timme:HomotopyContinuation.jl: A package for homotopy continuation in Julia, Lecture Notes in Computer Science10931, 458-465, 2018. · Zbl 1396.14003
[8] N. Bruin and E. Sert¨oz:Prym varieties of genus four curves, Transactions of the American Mathematical Society (2020),arXiv:1808.07881.
[9] M. Brundu and A. Logar:Parametrization of the orbits of cubic surfaces, Transformation Groups3(1998) 209-239. · Zbl 0938.14029
[10] A. Buckley and T. Ko´sir:Determinantal representations of smooth cubic surfaces 125(2007) 115-140.
[11] M. Chan and P. Jiradilok:Theta characteristics of tropical K4-curves, Combinatorial Algebraic Geometry, 65-86, Fields Inst. Comm.80, Springer, New York. · Zbl 1390.14195
[12] A. Clebsch:Die Geometrie auf den Fl¨achen dritter Ordnung, J. Reine Angew. Math.65(1866) 359-380.
[13] L. Cremona:M´emoire de g´eom´etrie pure sur les surfaces du troisi´eme ordre, Journal des Math. pures et appl.68(1868)
[14] M.A. Cueto, A. Deopurkar:Anticanonical tropical cubic del Pezzos contain exactly 27 lines,arXiv:1906.08196
[15] H. Derksen and G. Kemper:Computational Invariant Theory, Encyclopedia of Mathematical Sciences, vol 130, Springer-Verlag, Berlin 2002. · Zbl 1011.13003
[16] I. V. Dolgachev:Luigi Cremona and cubic surfaces. In Luigi Cremona (18301903). Convegno di Studi matematici, Istituto Lombardo, Accademia di Scienze e Lettere, Milano (2005), 55-70
[17] I.V. Dolgachev:Classical Algebraic Geometry, A Modern View, Cambridge University Press, 2012. · Zbl 1252.14001
[18] I.V. Dolgachev, B. van Geemen, S. Kondo:A complex ball uniformization of the moduli space of cubic surfaces via periods of K3 surfaces, J. Reine Angew. Math. 588(2005), 99-148. · Zbl 1090.14010
[19] F.E. Eckardt:Uber diejenigen Fl¨achen dritten Grades, auf denen sich drei gerade Linien in einem Punkte schneiden, Mathematische Annalen10(1876) 227-272 · JFM 08.0510.02
[20] W.L. Edge:The discriminant of a cubic surface, Proceedings of the Royal Irish Academy Ser. A80A(1980) 75-78. · Zbl 0436.51018
[21] C. Geiser:Uber die Doppeltangenten einer ebenen Curve vierten Grades¨, Mathematische Annalen1(1860) 129-138. · JFM 02.0417.01
[22] I. M. Gel’fand, M. Kapranov and A. Zelevinsky:Discriminants, Resultants, and Multidimensional Determinants, Birkh¨auser, Boston, MA, 1994.
[23] P. Griffiths and J. Harris:Principles of Algebraic Geometry, Wiley & Sons, 1978. · Zbl 0408.14001
[24] M. Gross, P. Hacking, S. Keel, B. Siebert:The mirror of the cubic surface, arXiv:1910.08427
[25] P. Hacking, S. Keel and J. Tevelev:Stable pair, tropical, and log canonical compactifications of moduli spaces of del Pezzo surfaces, Invent. Math.178(2009) 173-227. · Zbl 1205.14012
[26] R. Hartshorne:Algebraic Geoometry, GTM52, Springer (1977)
[27] C. Jordan:Trait´e des substitutions et ´equations alg´ebriques, Paris, Gauthier- Villars, 1870.
[28] M. Joswig, M. Panizzut and B. Sturmfels:The Schl¨afli fan,arXiv:1905.11951. · Zbl 1505.14128
[29] B.Ya. Kazarnovskii:Newton polyhedra and the B´ezout formula for matrix-valued functions of finite-dimensional representations, Functional Analysis and its Applications21(1987) 319-321.
[30] J. Koll´ar:Real algebraic surfaces, Lecture notes of Trento summer school, September 1997,arXiv:alg-geom/9712003.
[31] D. Maclagan and B. Sturmfels:Introduction to Tropical Geometry, Graduate Studies in Mathematics, Vol 161, American Mathematical Society, 2015. · Zbl 1321.14048
[32] M. Michałek and H. Moon:Spaces of sums of powers and real rank boundaries, Beitr¨age zur Algebra und Geometrie59(2018) 645-663. · Zbl 1403.14090
[33] E. J. Nansen:On the eliminant of a set of quadrics, ternary or quaternary, Proceedings of the Royal Society of Edinburgh22(1899) 353-358. · JFM 30.0161.09
[34] M. Panizzut and M.D. Vigeland:Tropical Lines on Cubic Surfaces, arXiv:0708.3847(revised 2019).
[35] I. Polo-Blanco and J. Top:Explicit real cubic surfaces, Canadian Mathematical Bulletin11(2008) 125-133. · Zbl 1132.14333
[36] M. Reid:Undergraduate Algebraic Geometry, LMS Student texts12, Cambridge University Press, 1988.
[37] Q. Ren, K. Shaw and B. Sturmfels:Tropicalization of del Pezzo surfaces, Advances in Mathematics300(2016) 156-189. · Zbl 1375.14218
[38] G. Salmon:On quaternary cubics, Philosophical Transactions of the Royal Society,150(1861) 229-239.
[39] L. Schl¨afli:An attempt to determine the twenty-seven lines upon a surface of the third order and to divide such surfaces into species in reference to the reality of the lines upon the surface, Quart. J. Math.2(1858) 55-65, 110-121.
[40] L. Schl¨afli:On the distribution of surfaces of the third order into species, in reference to the absence or presence of singular points, and the reality of their lines, Phil. Trans. of Roy. Soc. London6(1863) 201-241.
[41] B. Segre:The Non-singular Cubic Surfaces, Oxford University Press, 1942 · JFM 68.0358.01
[42] C. Segre:Sur la g´en´eration projective des surfaces cubiques, Archiv der Math. und Phys.10(1906) 209-215 · JFM 37.0576.02
[43] A. Seigal:Ranks and symmetric ranks of cubic surfaces,arXiv:1801.05377. · Zbl 1444.14091
[44] J. Steiner:Uber die Fl¨achen dritten Grades¨, Journ. f¨ur reiner und angew. Math., 53(1856)e 133-141
[45] B. Sturmfels and Z. Xu:Sagbi bases of Cox-Nagata rings, Journal of the European Mathematical Society12(2010) 429-459. · Zbl 1202.14053
[46] I. Vainsencher:Hypersurfaces with up to six double points, Communications in Algebra31(2003) 4107-4129. · Zbl 1061.14059
[47] M.D. Vigeland:Smooth tropical surfaces with infinitely many tropical lines, Ark. Mat. 48 (2010),1, 177-206. · Zbl 1198.14061
[48] M. Brandt and A. Geiger:A tropical count of binodal cubic surfaces, Le Matematiche, this volume.
[49] L. Brustenga, S. Timme, M. Weinstein:96120: The degree of the linear orbit of a cubic surface, Le Matematiche, this volume.
[50] D. Bunnett and H. Keneshlou:Determinantal representations of the cubic discriminant, Le Matematiche, this volume.
[51] E. Cazzador and B. Skauli:Towards the degree of the PGL(4)-orbit of a cubic surface, Le Matematiche, this volume.
[52] T. C¸ elik, F. Galuppi, A. Kulkarni and M.-S. Sorea:On the eigenpoints of cubic surfaces, Le Matematiche, this volume.
[53] R. Dinu and T. Seynnaeve:The Hessian discriminant, Le Matematiche, this volume.
[54] M. Donten-Bury, P. G¨orlach and M. Wrobel:Towards classifying toric degenerations of cubic surfaces, Le Matematiche, this volume.
[55] A.-S. Elsenhans and J. Jahnel:Computing invariants of cubic surfaces, Le Matematiche, this volume.
[56] A. Geiger:On realizability of lines on tropical cubic surfaces and the BrunduLogar normal form, Le Matematiche, this volume.
[57] M. Hahn, S. Lamboglia and A. Vargas:A short note on Cayley-Salmon equations, Le Matematiche, this volume.
[58] K. Kastner and R. L¨owe:The Newton polytope of the discriminant of a quaternary 424KRISTIAN RANESTAD - BERND STURMFELS
[59] H. Keneshlou:Cubic surfaces on the singular locus of the Eckardt hypersurface, Le Matematiche, this volume.
[60] M. Panizzut, E. Sert¨oz and B. Sturmfels:An octanomial model for cubic surfaces, Le Matematiche, this volume.
[61] A.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.