LLR2InventionesPublished

LLR2InventionesPublished - DOI: 10.1007/s00222-004-0403-2...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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: lorenzini@math.uga.edu ) 3 Universit de Paris-Sud, Bt. 425, 91405 Orsay Cedex, France (e-mail: michel.raynaud@math.u-psud.fr ) 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 [15], 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 [16], remark after 1.17, showed, improving on a result of Milne [9], that the Brauer group of a rational surface has order a square. In [17], 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 ?...
View Full Document

Page1 / 4

LLR2InventionesPublished - DOI: 10.1007/s00222-004-0403-2...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online