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The Brauer‐Manin obstruction on Del Pezzo surfaces of degree 2
Author(s) -
Corn Patrick
Publication year - 2007
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/pdm015
Subject(s) - mathematics , degree (music) , brauer group , conjecture , computation , abelian group , hasse principle , pure mathematics , diagonal , class (philosophy) , algebraic number field , geometry , algorithm , computer science , physics , artificial intelligence , acoustics
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 H 1 ( 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.