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Proceedings of the London Mathematical Society Advance Access originally published online on June 7, 2007
Proceedings of the London Mathematical Society 2007 95(3):735-777; doi:10.1112/plms/pdm015
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© 2007 London Mathematical Society

The Brauer-Manin obstruction on Del Pezzo surfaces of degree 2

Patrick Corn

Department of Mathematics
University of Georgia
Athens, GA 30602-7403
USA
http://www.math.uga.edu/~corn

Received 10 April 2006. Revision received 19 December 2006.

This paper explores the computation of the Brauer-Manin obstruction on Del Pezzo surfaces of degree 2, with examples coming from the class of ‘semi-diagonal’ Del Pezzo surfaces of degree 2. It is conjectured that the failure of the Hasse principle for a broad class of varieties, including Del Pezzo surfaces, can always be explained by a non-trivial Brauer-Manin obstruction. We provide computational evidence in support of this conjecture for semi-diagonal Del Pezzo surfaces of degree 2. In addition, we determine the complete list of the possibilities for the finite abelian group H1 (k, Pic X), where X is a Del Pezzo surface of any degree, thus completing a computation which had been previously carried out in various special cases only.

2000 Mathematics Subject Classification 14G05.


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