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Unformatted text preview: DOI: 10.1007/s00222-004-0403-2 Invent. math. 159, 673676 (2005) On the Brauer group of a surface Qing Liu 1 , Dino Lorenzini 2 ,? , Michel Raynaud 3 1 CNRS, Laboratoire A2X, Universit de Bordeaux I, 33405 Talence, France (e-mail: Qing.Liu@math.u-bordeaux1.fr ) 2 Department of Mathematics, University of Georgia, Athens, GA 30602, USA (e-mail: firstname.lastname@example.org ) 3 Universit de Paris-Sud, Bt. 425, 91405 Orsay Cedex, France (e-mail: email@example.com ) Oblatum 19-IV-2004 & 10-VIII-2004 Published online: 22 December 2004 Springer-Verlag 2004 Siegfried Bosch zum 60. Geburtstag gewidmet Our goal in this note is to complete the proof of the following theorem. Theorem 1. Let k be a finite field, of characteristic p. Let X / k be a smooth proper geometrically connected surface. Assume that for some prime ` , the `-part of the group Br ( X ) is finite. Then | Br ( X ) | is a square. Artin and Tate , 5.1, have shown in 1966 the existence of a canonical skew-symmetric pairing on the non- p part of Br ( X ) , whose kernel is exactly the set of divisible elements. It follows from this fact that if the non- p part of Br ( X ) is finite, then its order is a square or twice a square. A few years later, Manin published examples of rational surfaces (that is, surfaces birational over k to the projective plane) with Brauer groups equal to Z / 2 Z . It is only in 1996 that the examples of Manin were revisited by Urabe, who found a mistake in them. For rational surfaces, the Brauer group is relatively easy to understand, and Urabe , remark after 1.17, showed, improving on a result of Milne , that the Brauer group of a rational surface has order a square. In , 0.3, Urabe then proves in full generality that when p 6= 2, the 2-part of Br ( X ) modulo its divisible subgroup has order a square. Thus, to complete the proof of Theorem 1, it remains to treat the case where p = 2 and X / k is not rational. As we remarked above, it was wrongly assumed for about 30 years that | Br ( X ) | was not always a square. Our method of proof provides for any p that the 2-part of Br ( X ) has order a square via the knowledge of the 2-part ?...
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