no. 3
A family of varieties with exactly one pointless rational fiber
Viray, Bianca1 
1 Mathematics Department Box 1917 Brown University Providence, RI 02912 USA
Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 741-745

We construct a concrete example of a 1-parameter family of smooth projective geometrically integral varieties over an open subscheme of ℙ ℚ 1 such that there is exactly one rational fiber with no rational points. This makes explicit a construction of Poonen.

On construit un exemple concret d’une famille à un paramètre de variétés lisses, projectives, et géométriquement intègres sur un sous-schéma ouvert de ℙ ℚ 1 , de sorte qu’il y ait précisément une fibre rationnelle sans point rationnel. Ceci rend explicite une construction de Poonen.

DOI : 10.5802/jtnb.743

Viray, Bianca 1

1 Mathematics Department Box 1917 Brown University Providence, RI 02912 USA 
@article{JTNB_2010__22_3_741_0,
     author = {Viray, Bianca},
     title = {A family of varieties with exactly one pointless rational fiber},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {741--745},
     year = {2010},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {3},
     doi = {10.5802/jtnb.743},
     zbl = {1222.14040},
     mrnumber = {2769342},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/jtnb.743/}
}
TY  - JOUR
AU  - Viray, Bianca
TI  - A family of varieties with exactly one pointless rational fiber
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2010
SP  - 741
EP  - 745
VL  - 22
IS  - 3
PB  - Université Bordeaux 1
UR  - https://www.numdam.org/articles/10.5802/jtnb.743/
DO  - 10.5802/jtnb.743
LA  - en
ID  - JTNB_2010__22_3_741_0
ER  - 
%0 Journal Article
%A Viray, Bianca
%T A family of varieties with exactly one pointless rational fiber
%J Journal de théorie des nombres de Bordeaux
%D 2010
%P 741-745
%V 22
%N 3
%I Université Bordeaux 1
%U https://www.numdam.org/articles/10.5802/jtnb.743/
%R 10.5802/jtnb.743
%G en
%F JTNB_2010__22_3_741_0
Viray, Bianca. A family of varieties with exactly one pointless rational fiber. Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 3, pp. 741-745. doi: 10.5802/jtnb.743

[1] Wieb Bosma, John Cannon, Catherine Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265. | Zbl | MR

[2] Henri Cohen Number theory. Vol. I. Tools and Diophantine equations Springer, 2007. | Zbl | MR

[3] Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, Peter Swinnerton-Dyer Intersections of two quadrics and Châtelet surfaces. I. J. Reine Angew. Math. 373 (1987), 37–107. | Zbl | MR

[4] Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, Peter Swinnerton-Dyer Intersections of two quadrics and Châtelet surfaces. II. J. Reine Angew. Math. 374 (1987), 72–168. | Zbl | MR

[5] V.A. Iskovskih A counterexample to the Hasse principle for systems of two quadratic forms in five variables Mat. Zametki 10 (1971), 253–257 | Zbl | MR

[6] Bjorn Poonen Existence of rational points on smooth projective varieties J. Eur. Math. Soc. (JEMS) 3 529–543 | Zbl | MR

Cité par Sources :