The arithmetic of certain del Pezzo surfaces and K3 surfaces

Dong Quan Ngoc Nguyen[1]

  • [1] Department of Mathematics The University of Arizona, 617 N. Santa Rita Ave, Tucson, Arizona 85721, USA

Journal de Théorie des Nombres de Bordeaux (2012)

  • Volume: 24, Issue: 2, page 447-460
  • ISSN: 1246-7405

Abstract

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We construct del Pezzo surfaces of degree 4 violating the Hasse principle explained by the Brauer-Manin obstruction. Using these del Pezzo surfaces, we show that there are algebraic families of K 3 surfaces violating the Hasse principle explained by the Brauer-Manin obstruction. Various examples are given.

How to cite

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Nguyen, Dong Quan Ngoc. "The arithmetic of certain del Pezzo surfaces and K3 surfaces." Journal de Théorie des Nombres de Bordeaux 24.2 (2012): 447-460. <http://eudml.org/doc/251067>.

@article{Nguyen2012,
abstract = {We construct del Pezzo surfaces of degree $4$ violating the Hasse principle explained by the Brauer-Manin obstruction. Using these del Pezzo surfaces, we show that there are algebraic families of $K3$ surfaces violating the Hasse principle explained by the Brauer-Manin obstruction. Various examples are given.},
affiliation = {Department of Mathematics The University of Arizona, 617 N. Santa Rita Ave, Tucson, Arizona 85721, USA},
author = {Nguyen, Dong Quan Ngoc},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Hasse principle; degree 4 del Pezzo surfaces; surfaces; Brauer-Manin obstruction},
language = {eng},
month = {6},
number = {2},
pages = {447-460},
publisher = {Société Arithmétique de Bordeaux},
title = {The arithmetic of certain del Pezzo surfaces and K3 surfaces},
url = {http://eudml.org/doc/251067},
volume = {24},
year = {2012},
}

TY - JOUR
AU - Nguyen, Dong Quan Ngoc
TI - The arithmetic of certain del Pezzo surfaces and K3 surfaces
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2012/6//
PB - Société Arithmétique de Bordeaux
VL - 24
IS - 2
SP - 447
EP - 460
AB - We construct del Pezzo surfaces of degree $4$ violating the Hasse principle explained by the Brauer-Manin obstruction. Using these del Pezzo surfaces, we show that there are algebraic families of $K3$ surfaces violating the Hasse principle explained by the Brauer-Manin obstruction. Various examples are given.
LA - eng
KW - Hasse principle; degree 4 del Pezzo surfaces; surfaces; Brauer-Manin obstruction
UR - http://eudml.org/doc/251067
ER -

References

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  10. Yu.I. Manin, Le groupe de Brauer-Grothendieck en géométrie diophantienne. In Actes du Congrès International des Mathématiciens, (Nice, 1970), pp. 401–411. Zbl0239.14010MR427322
  11. B. Poonen, An explicit family of genus-one curves violating the Hasse principle. J. Théor. Nombres Bordeaux 13 (2001), no. 1, 263–274. Zbl1046.11038MR1838086
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  13. H. Reichardt, Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen. J. Reine Angew. Math. 184 (1942), pp. 12–18. Zbl68.0070.01MR9381
  14. M. Reid, The complete intersection of two or more quadrics. Ph.D thesis, University of Cambridge, 1972. 
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