A family of varieties with exactly one pointless rational fiber

Bianca Viray[1]

  • [1] Mathematics Department Box 1917 Brown University Providence, RI 02912 USA

Journal de Théorie des Nombres de Bordeaux (2010)

  • Volume: 22, Issue: 3, page 741-745
  • ISSN: 1246-7405

Abstract

top
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.

How to cite

top

Viray, Bianca. "A family of varieties with exactly one pointless rational fiber." Journal de Théorie des Nombres de Bordeaux 22.3 (2010): 741-745. <http://eudml.org/doc/116431>.

@article{Viray2010,
abstract = {We construct a concrete example of a $1$-parameter family of smooth projective geometrically integral varieties over an open subscheme of $\mathbb\{P\}^1_\{\mathbb\{Q\}\}$ such that there is exactly one rational fiber with no rational points. This makes explicit a construction of Poonen.},
affiliation = {Mathematics Department Box 1917 Brown University Providence, RI 02912 USA},
author = {Viray, Bianca},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {3},
pages = {741-745},
publisher = {Université Bordeaux 1},
title = {A family of varieties with exactly one pointless rational fiber},
url = {http://eudml.org/doc/116431},
volume = {22},
year = {2010},
}

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
PB - Université Bordeaux 1
VL - 22
IS - 3
SP - 741
EP - 745
AB - We construct a concrete example of a $1$-parameter family of smooth projective geometrically integral varieties over an open subscheme of $\mathbb{P}^1_{\mathbb{Q}}$ such that there is exactly one rational fiber with no rational points. This makes explicit a construction of Poonen.
LA - eng
UR - http://eudml.org/doc/116431
ER -

References

top
  1. Wieb Bosma, John Cannon, Catherine Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265. Zbl0898.68039MR1484478
  2. Henri CohenNumber theory. Vol. I. Tools and Diophantine equations Springer, 2007. Zbl1119.11001MR2312337
  3. Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, Peter Swinnerton-DyerIntersections of two quadrics and Châtelet surfaces. I. J. Reine Angew. Math. 373 (1987), 37–107. Zbl0622.14029MR870307
  4. Jean-Louis Colliot-Thélène, Jean-Jacques Sansuc, Peter Swinnerton-DyerIntersections of two quadrics and Châtelet surfaces. II. J. Reine Angew. Math. 374 (1987), 72–168. Zbl0622.14030MR876222
  5. V.A. IskovskihA counterexample to the Hasse principle for systems of two quadratic forms in five variables Mat. Zametki 10 (1971), 253–257 Zbl0221.10028MR286743
  6. Bjorn PoonenExistence of rational points on smooth projective varieties J. Eur. Math. Soc. (JEMS) 3 529–543 Zbl1183.14032MR2505440

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.