## On the Néron-Severi group of surfaces with many lines

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- by Samuel Boissière and Alessandra Sarti PDF
- Proc. Amer. Math. Soc.
**136**(2008), 3861-3867 Request permission

## 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

- 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** - Hiroshi Inose,
*On certain Kummer surfaces which can be realized as non-singular quartic surfaces in $P^{3}$*, J. Fac. Sci. Univ. Tokyo Sect. IA Math.**23**(1976), no. 3, 545–560. MR**429915** - Masato Kuwata,
*Elliptic fibrations on quartic $K3$ surfaces with large Picard numbers*, Pacific J. Math.**171**(1995), no. 1, 231–243. MR**1362985**, DOI 10.2140/pjm.1995.171.231 - 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. - I. I. Pjateckiĭ-Šapiro and I. R. Šafarevič,
*Torelli’s theorem for algebraic surfaces of type $\textrm {K}3$*, Izv. Akad. Nauk SSSR Ser. Mat.**35**(1971), 530–572 (Russian). MR**0284440** - Nobuo Sasakura,
*On some results on the Picard numbers of certain algebraic surfaces*, J. Math. Soc. Japan**20**(1968), 297–321. MR**228495**, DOI 10.2969/jmsj/02010297 - B. Segre,
*On arithmetical properties of quartic surfaces*, Proc. London Math. Soc. (2)**49**(1947), 353–395. MR**21952**, DOI 10.1112/plms/s2-49.5.353 - 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**, DOI 10.24033/asens.1407 - T. Shioda and H. Inose,
*On singular $K3$ surfaces*, Complex analysis and algebraic geometry, Iwanami Shoten, Tokyo, 1977, pp. 119–136. MR**0441982** - Tetsuji Shioda and Naoki Mitani,
*Singular abelian surfaces and binary quadratic forms*, Classification of algebraic varieties and compact complex manifolds, Lecture Notes in Math., Vol. 412, Springer, Berlin, 1974, pp. 259–287. MR**0382289** - Joseph H. Silverman,
*Advanced topics in the arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR**1312368**, DOI 10.1007/978-1-4612-0851-8

## Additional Information

**Samuel Boissière**- Affiliation: Laboratoire J.A. Dieudonné UMR CNRS 6621, Université de Nice Sophia-Antipolis, Parc Valrose, 06108 Nice, France
- Email: samuel.boissiere@math.unice.fr
**Alessandra Sarti**- Affiliation: Johannes Gutenberg Universität Mainz, Institut für Mathematik, 55099 Mainz, Germany
- MR Author ID: 651260
- Email: sarti@mathematik.uni-mainz.de
- 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
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**136**(2008), 3861-3867 - MSC (2000): Primary 14J28
- DOI: https://doi.org/10.1090/S0002-9939-08-09475-6
- MathSciNet review: 2425725