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.

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

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

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.743
@article{JTNB_2010__22_3_741_0,
     author = {Bianca Viray},
     title = {A family of varieties with exactly one pointless rational fiber},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {741--745},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {3},
     year = {2010},
     doi = {10.5802/jtnb.743},
     zbl = {1222.14040},
     mrnumber = {2769342},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.743/}
}
Bianca Viray. 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. https://jtnb.centre-mersenne.org/articles/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. | MR 1484478 | Zbl 0898.68039

[2] Henri Cohen Number theory. Vol. I. Tools and Diophantine equations Springer, 2007. | MR 2312337 | Zbl 1119.11001

[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. | MR 870307 | Zbl 0622.14029

[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. | MR 876222 | Zbl 0622.14030

[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 | MR 286743 | Zbl 0221.10028

[6] Bjorn Poonen Existence of rational points on smooth projective varieties J. Eur. Math. Soc. (JEMS) 3 529–543 | MR 2505440 | Zbl 1183.14032