Hostname: page-component-8448b6f56d-jr42d Total loading time: 0 Render date: 2024-04-16T16:53:53.759Z Has data issue: false hasContentIssue false
  • English
  • Français

CONICS IN SEXTIC $K3$-SURFACES IN $\mathbb {P}^4$

Part of: Curves Projective and enumerative geometry Surfaces and higher-dimensional varieties

Published online by Cambridge University Press:  29 November 2021

ALEX DEGTYAREV Open the ORCID record for ALEX DEGTYAREV [Opens in a new window]
ALEX DEGTYAREV*
Affiliation:
Department of Mathematics Bilkent University06800Ankara, Turkeydegt@fen.bilkent.edu.tr
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that the maximal number of conics in a smooth sextic $K3$ -surface $X\subset \mathbb {P}^4$ is 285, whereas the maximal number of real conics in a real sextic is 261. In both extremal configurations, all conics are irreducible.

MSC classification

Primary: 14J28: $K3$ surfaces and Enriques surfaces
Secondary: 14H50: Plane and space curves 14N20: Configurations and arrangements of linear subspaces 14N25: Varieties of low degree
Type
Article
Information
Nagoya Mathematical Journal , Volume 246 , June 2022 , pp. 273 - 304
Check for updates
Copyright
© (2021) The Authors. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author was partially supported by the TÜBİTAK grant 118F413.

References

Barth, W. and Bauer, T., Smooth quartic surfaces with 352 conics, Manuscripta Math. 85 (1994), 409–417.10.1007/BF02568207CrossRefGoogle Scholar
Bauer, T., Quartic surfaces with 16 skew conics, J. Reine Angew. Math. 464 (1995), 207–217.Google Scholar
Cayley, A., On the triple tangent planes of surfaces of the third order, Cambridge and Dublin Math. J. 4 (1849), 118–138.Google Scholar
Conway, J. H. and Sloane, N. J. A., Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 290, Springer, New York, 1988, With contributions by E. Bannai, J. Leech, S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen, and B. B. Venkov.10.1007/978-1-4757-2016-7CrossRefGoogle Scholar
Degtyarev, A., Lines on smooth polarized K3-surfaces, Discrete Comput. Geom. 62 (2019), 601–648.10.1007/s00454-018-0038-5CrossRefGoogle Scholar
Degtyarev, A., Smooth models of singular $K3$ -surfaces, Rev. Mat. Iberoam. 35 (2019), 125–172.10.4171/rmi/1051CrossRefGoogle Scholar
Degtyarev, A., Tritangents to smooth sextic curves, to appear in Ann. Inst. Fourier., preprint, arXiv:1909.05657 [math.AG].Google Scholar
Degtyarev, A., Itenberg, I., and Ottem, J. C., Planes in cubic four folds, preprint, arXiv:2105.13951.Google Scholar
Degtyarev, A., Itenberg, I., and Sertöz, A. S., Lines on quartic surfaces, Math. Ann. 368 (2017), 753–809.10.1007/s00208-016-1484-0CrossRefGoogle Scholar
Elkies, N. D., “Upper bounds on the number of lines on a surface” in New Trends in Arithmetic and Geometry of Algebraic Surfaces, BIRS Conference 17w5146, Banff International Research Station for Mathematical Innovation and Discovery, 2017, https://open.library.ubc.ca/cIRcle/collections/48630/items/1.0355541.Google Scholar
GAP, GAP—Groups, Algorithms, and Programming, Version 4.10.1, February 2019, https://www.gap-system.org.Google Scholar
Güneş Aktaş, Ç., Real representatives of equisingular strata of simple quartic surfaces, Internat. J. Math. 30 (2019).10.1142/S0129167X19500630CrossRefGoogle Scholar
Huybrechts, D., Lectures on K3 Surfaces, Cambridge Studies in Adv. Math. 158, Cambridge University Press, Cambridge, MA, 2016.10.1017/CBO9781316594193CrossRefGoogle Scholar
Kulikov, V. S., Surjectivity of the period mapping for $K3$ surfaces, Uspehi Mat. Nauk 32 (1977), 257–258.Google Scholar
McKay, B. D., Nauty user’s guide (Version 1.5), Technical report TR-CS-90-0, Computer Science Department, Australian National University, 1990.Google Scholar
McKay, B. D. and Piperno, A., Practical graph isomorphism, II, J. Symbolic Comput. 60 (2014), 94–112.10.1016/j.jsc.2013.09.003CrossRefGoogle Scholar
Miyaoka, Y., Counting lines and conics on a surface, Publ. Res. Inst. Math. Sci. 45 (2009), 919–923.10.2977/prims/1249478969CrossRefGoogle Scholar
Niemeier, H.-V., Definite quadratische Formen der Dimension  $24$  und Diskriminante  $1$ , J. Number Theory 5 (1973), 142–178.10.1016/0022-314X(73)90068-1CrossRefGoogle Scholar
Nikulin, V. V., Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), 111–177, 238, English translation: Math USSR-Izv. 14 (1979), 103–167 (1980).Google Scholar
Nikulin, V. V., “Weil linear systems on singular K3 surfaces ” in Algebraic Geometry and Analytic Geometry (Tokyo, 1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, 138–164.Google Scholar
Pjateckiĭ-Šapiro, I. I. and Šafarevič, I. R., Torelli’s theorem for algebraic surfaces of type K3, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572, English translation: Math. USSR-Izv. 5 (1971), 547–588.Google Scholar
Rams, S. and Schütt, M., 64 lines on smooth quartic surfaces, Math. Ann. 362 (2015), 679–698.10.1007/s00208-014-1139-yCrossRefGoogle Scholar
Rams, S. and Schütt, M., Counting lines on surfaces, especially quintics, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 20 (2020),859–890.Google Scholar
Rams, S. and Schütt, M., 24 rational curves on K3 surfaces, preprint, arXiv:1907.04182 [math.AG].Google Scholar
Saint-Donat, B., Projective models of  $K\kern-2pt$ - $3$ surfaces, Amer. J. Math. 96 (1974), 602–639.10.2307/2373709CrossRefGoogle Scholar
Schur, F., Ueber eine besondre Classe von Flächen vierter Ordnung, Math. Ann. 20 (1882), 254–296.10.1007/BF01446525CrossRefGoogle Scholar
Schütt, M., Fields of definition of singular  $K3$  surfaces, Commun. Number Theory Phys. 1 (2007), 307–321.10.4310/CNTP.2007.v1.n2.a2CrossRefGoogle Scholar
Segre, B., The maximum number of lines lying on a quartic surface, Quart. J. Math., Oxford Ser. 14 (1943), 86–96.10.1093/qmath/os-14.1.86CrossRefGoogle Scholar
Shimada, I., “Non-homeomorphic conjugate complex varieties ” in Singularities—Niigata–Toyama 2007, Adv. Stud. Pure Math. 56, Math. Soc. Japan, Tokyo, 2009, 285–301.10.2969/aspm/05610285CrossRefGoogle Scholar
Soicher, L. H., GRAPE, GRaph Algorithms using PErmutation groups, Version 4.8.1, October 2018, Refereed GAP package, https://gap-packages.github.io/grape.Google Scholar