Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the Néron-Severi group of surfaces with many lines

Authors: Samuel Boissière and Alessandra Sarti
Journal: Proc. Amer. Math. Soc. 136 (2008), 3861-3867
MSC (2000): Primary 14J28
Published electronically: June 3, 2008
MathSciNet review: 2425725
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a binary quartic form $ \phi$ without multiple factors, we classify the quartic $ K3$ surfaces $ \phi(x,y)=\phi(z,t)$ whose Néron-Severi group is (rationally) generated by lines. For generic binary forms $ \phi$, $ \psi$ of prime degree without multiple factors, we prove that the Néron-Severi group of the surface $ \phi(x,y)=\psi(z,t)$ is rationally generated by lines.

References [Enhancements On Off] (What's this?)

  • [BS07] Samuel Boissière and Alessandra Sarti, Counting lines on surfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 6 (2007), no. 1, 39–52. MR 2341513
  • [Ino76] Hiroshi Inose, On certain Kummer surfaces which can be realized as non-singular quartic surfaces in 𝑃³, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 23 (1976), no. 3, 545–560. MR 0429915
  • [Kuw95] Masato Kuwata, Elliptic fibrations on quartic 𝐾3 surfaces with large Picard numbers, Pacific J. Math. 171 (1995), no. 1, 231–243. MR 1362985
  • [Ogu] Keiji Oguiso, Picard numbers in a family of hyperkähler manifolds - A supplement to the article of R. Borcherds, L. Katzarkov, T. Pantev, N. I. Shepherd-Barron, arXiv:math.AG/0011258.
  • [PŠŠ71] I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič, Torelli’s theorem for algebraic surfaces of type 𝐾3, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 530–572 (Russian). MR 0284440
  • [Sas68] Nobuo Sasakura, On some results on the Picard numbers of certain algebraic surfaces, J. Math. Soc. Japan 20 (1968), 297–321. MR 0228495
  • [Seg47] B. Segre, On arithmetical properties of quartic surfaces, Proc. London Math. Soc. (2) 49 (1947), 353–395. MR 0021952
  • [Shi81] Tetsuji Shioda, On the Picard number of a complex projective variety, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 303–321. MR 644520
  • [SI77] T. Shioda and H. Inose, On singular 𝐾3 surfaces, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136. MR 0441982
  • [SM74] Tetsuji Shioda and Naoki Mitani, Singular abelian surfaces and binary quadratic forms, Classification of algebraic varieties and compact complex manifolds, Springer, Berlin, 1974, pp. 259–287. Lecture Notes in Math., Vol. 412. MR 0382289
  • [Sil94] Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 14J28

Retrieve articles in all journals with MSC (2000): 14J28

Additional Information

Samuel Boissière
Affiliation: Laboratoire J.A. Dieudonné UMR CNRS 6621, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice, France

Alessandra Sarti
Affiliation: Johannes Gutenberg Universität Mainz, Institut für Mathematik, 55099 Mainz, Germany

Keywords: N{\'e}ron-Severi group, Picard number, lines on surfaces
Received by editor(s): January 22, 2007
Received by editor(s) in revised form: March 29, 2007, and October 9, 2007
Published electronically: June 3, 2008
Communicated by: Ted Chinburg
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.