This paper deals with surfaces with many lines. It is well-known that a cubic contains of them and that the maximal number for a quartic is . In higher degree the question remains open. Here we study classical and new constructions of surfaces with high number of lines. We obtain a symmetric octic with lines, and give examples of surfaces of degree containing a sequence of skew lines.
@article{ASNSP_2007_5_6_1_39_0, author = {Boissi\`ere, Samuel and Sarti, Alessandra}, title = {Counting lines on surfaces}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {39--52}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 6}, number = {1}, year = {2007}, mrnumber = {2341513}, zbl = {1150.14013}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2007_5_6_1_39_0/} }
TY - JOUR AU - Boissière, Samuel AU - Sarti, Alessandra TI - Counting lines on surfaces JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2007 SP - 39 EP - 52 VL - 6 IS - 1 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2007_5_6_1_39_0/ LA - en ID - ASNSP_2007_5_6_1_39_0 ER -
%0 Journal Article %A Boissière, Samuel %A Sarti, Alessandra %T Counting lines on surfaces %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2007 %P 39-52 %V 6 %N 1 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2007_5_6_1_39_0/ %G en %F ASNSP_2007_5_6_1_39_0
Boissière, Samuel; Sarti, Alessandra. Counting lines on surfaces. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 6 (2007) no. 1, pp. 39-52. http://www.numdam.org/item/ASNSP_2007_5_6_1_39_0/
[1] Foundations of the theory of Fano schemes, Compositio Math. 34 (1977), 3-47. | Numdam | MR | Zbl
and ,[2] Fano varieties of lines on hypersurfaces, Arch. Math. (Basel) 31 (1978/79), 96-104. | MR | Zbl
and ,[3] How many rational points can a curve have?, In: “The moduli space of curves” (Texel Island, 1994), Progr. Math., Vol. 129, Birkhäuser Boston, Boston, MA, 1995, 13-31. | MR | Zbl
, and ,[4] Singular 2.0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern, 2001, http://www.singular.uni-kl.de. | Zbl
, and ,[5] “Algebraic geometry”, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York, 1977. | MR | Zbl
,[6] “Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade”, Birkhäuser Verlag, Basel, 1993, Reprint of the 1884 original, Edited, with an introduction and commentary by Peter Slodowy. | JFM | MR | Zbl
,[7] The maximal number of quotient singularities on surfaces with given numerical invariants, Math. Ann. 268 (1984), 159-171. | MR | Zbl
,[8] Kummer surfaces, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 278-293, 471. English translation: Math. USSR Izv. 9 (1975), 261-275. | MR | Zbl
,[9] Three-divisible families of skew lines on a smooth projective quintic, Trans. Amer. Math. Soc. 354 (2002), 2359-2367 (electronic). | MR | Zbl
,[10] Projective surfaces with many skew lines, Proc. Amer. Math. Soc. 133 (2005), 11-13 (electronic). | MR | Zbl
,[11] Pencils of symmetric surfaces in , J. Algebra 246 (2001), 429-452. | MR | Zbl
,[12] The maximum number of lines lying on a quartic surface, Quart. J. Math., Oxford Ser. 14 (1943), 86-96. | MR | Zbl
,[13] On arithmetical properties of quartic surfaces, Proc. London Math. Soc. (2) 49 (1947), 353-395. | MR | Zbl
,