Comptes Rendus
no. 5
Algebraic Geometry
Brauer obstruction for a universal vector bundle
[Obstruction de Brauer pour un fibré vectoriel universel]

Présenté par : Gérard Laumon

Vikraman Balaji1 ; Indranil Biswas2 ; Ofer Gabber3 ; Donihakkalu S. Nagaraj4
1 Chennai Mathematical Institute, Plot H1, SIPCOT IT Park, Padur, PO Siruseri 603103, India
2 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
3 Institut des Hautes Études Scientifiques, Le Bois-Marie, 35, route de Chartres, 91440 Bures-sur-Yvette, France
4 The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India
Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 265-268.

Soit X une courbe projective lisse de genre g(X)>2 et soit M l'espace de modules paramétrant les fibrés vectoriels E stables sur X de rang r et ayant déterminant ⋀rE=ξ, où ξ est un fibré en droites donné. Nous montrons que le groupe de Brauer Br(M) est égale à Z/nZ, où n=pgcd(r,degξ). De plus Br(M) est engendré par la classe du fibré projectif sur M de dimension relative r−1, obtenu par restriction du fibré projectif universel sur X×M en un point de X.

Let X be a smooth complex projective curve with genus(X)>2, and let M be the moduli space parametrizing isomorphism classes of stable vector bundles E over X of rank r with ⋀rE=ξ, where ξ is a fixed line bundle. We prove that the Brauer group Br(M) is Z/nZ, where n=g.c.d.(r,degree(ξ)). Moreover, Br(M) is generated by the class of the projective bundle over M of relative dimension r−1 obtained by restricting the universal projective bundle over X×M to a point of X.

Reçu le :
Accepté le :
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DOI : 10.1016/j.crma.2007.07.011

Vikraman Balaji 1 ; Indranil Biswas 2 ; Ofer Gabber 3 ; Donihakkalu S. Nagaraj 4

1 Chennai Mathematical Institute, Plot H1, SIPCOT IT Park, Padur, PO Siruseri 603103, India
2 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
3 Institut des Hautes Études Scientifiques, Le Bois-Marie, 35, route de Chartres, 91440 Bures-sur-Yvette, France
4 The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India 
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Vikraman Balaji; Indranil Biswas; Ofer Gabber; Donihakkalu S. Nagaraj. Brauer obstruction for a universal vector bundle. Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 265-268. doi : 10.1016/j.crma.2007.07.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.07.011/

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