Kresch, Andrew and Tschinkel, Yuri. (2004) On the arithmetic of del Pezzo surfaces of degree 2. Proceedings of the London Mathematical Society, Vol.89 (No.3). pp. 545-569. ISSN 0024-6115

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Abstract

We study the arithmetic of certain del Pezzo surfaces of degree 2. We produce examples of Brauer-Manin obstruction to the Hasse principle, coming from 2- and 4-torsion elements in the Brauer group.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Rational points (Geometry), Arithmetical algebraic geometry, Geometry, Algebraic, Galois cohomology, Surfaces, Algebraic
Journal or Publication Title:	Proceedings of the London Mathematical Society
Publisher:	Cambridge University Press
ISSN:	0024-6115
Date:	November 2004
Volume:	Vol.89
Number:	No.3
Page Range:	pp. 545-569
Identification Number:	10.1112/S002461150401490X
Status:	Peer Reviewed
Access rights to Published version:	Open Access
Funder:	Engineering and Physical Sciences Research Council (EPSRC), National Science Foundation (U.S.) (NSF)
Grant number:	0100277 (NSF)
References:	1. B. J. BIRCH and H. P. F. SWINNERTON-DYER, ‘The Hasse problem for rational surfaces’, J. Reine Angew. Math. 274/275 (1975) 164--174. 2. H. COHEN, Advanced topics in computational number theory (Springer, Berlin, 2000). 3. J.-L. COLLIOT-THÉLÈNE, ‘Intersections de deux quadriques et surfaces de Ch^atelet’, C. R. Acad. Sci. Paris S@er I Math. 298 (1984) 377--380. 4. J.-L. COLLIOT-THÉLÈNE, ‘Surfaces rationnelles ,brées en coniques de degré 4’, Séminaire de Théorie des Nombres, Paris 1988--1989 (ed. C. Goldstein), Progress in Mathematics 91 (Birkhäuser, Boston, 1990) 43--55. 5. J.-L. COLLIOT-THÉLÈNE, D. CORAY and J.-J. SANSUC, ‘Descente et principe de Hasse pour certaines variétés rationnelles’, J. Reine Angew. Math. 320 (1980) 150--191. 6. J.-L. COLLIOT-THÉLÈNE, D. KANEVSKY and J.-J. SANSUC, ‘Arithm@etique des surfaces cubiques diagonales’, Diophantine approximation and transcendence theory, Bonn, 1985 (ed. G. Wüstholz), Lecture Notes in Mathematics 1290 (Springer, Berlin, 1987) 1--108. 7. J.-L. COLLIOT-THÉLÈNE and J.-J. SANSUC, ‘Ladescente sur les variétés rationnelles’, Journées de Géometrie Algébrique d’Angers, July 1979 (ed. A. Beauville, NoordhoN, Alphen aan den Rijn, 1980) 223--237. 8. J.-L. COLLIOT-THÉLÈNE, J.-J. SANSUC and P. SWINNERTON-DYER, ‘Intersections of two quadrics and Châtelet surfaces’, J. Reine Angew. Math. 373 (1987) 37--107 and 374 (1987) 72--168. 9. C. FIEKER, ‘Über relative Normgleichungen in algebraischen Zahlkörpern’, PhD Thesis, Technische Universität Berlin, 1997. 10. C. FIEKER, A. JURK and M. POHST, ‘On solving relative norm equations in algebraic number ,elds’, Math. Comp. 66 (1997) 399--410. 11. V. A. ISKOVSKIH, ‘A counterexample to the Hasse principle for systems of two quadratic forms in ,ve variables’, Mat. Zametki 10 (1971) 253--257. 12. B. E. KUNYAVSKII, A. N. SKOROBOGATOV and M. A. TSFASMAN, ‘Del Pezzo surfaces of degree four’, M@em. Soc. Math. France 37 (1989). 13. YU. I. MANIN, Cubic forms, 2nd edn (North-Holland, Amsterdam, 1986). 14. P. SALBERGER, ‘Zero-cycles on rational surfaces over number ,elds’, Invent. Math. 91 (1988) 505--524. 15. C. L. SIEGEL, ‘Normen algebraischer Zahlen’, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1973) 197--215. 16. A. SKOROBOGATOV, Torsors and rational points (Cambridge University Press, 2001). 17. P. SWINNERTON-DYER, ‘Two special cubic surfaces’, Mathematika 9 (1962) 54--56. 18. P. SWINNERTON-DYER, ‘The Brauer group of cubic surfaces’, Math. Proc. Cambridge Philos. Soc. 113 (1993) 449--460. 19. P. SWINNERTON-DYER, ‘Rational points on some pencils of conics with 6 singular ,bres’, Ann. Fac. Sci. Toulouse Math. (6) 8 (1999) 331--341. 20. P. SWINNERTON-DYER, ‘The solubility of diagonal cubic surfaces’, Ann. Sci. @ Ecole Norm. Sup. (4) 34 (2001) 891--912.
URI:	http://wrap.warwick.ac.uk/id/eprint/757

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