Brauer obstruction for a universal vector bundle

Vikraman Balaji; Indranil Biswas; Ofer Gabber; Donihakkalu S. Nagaraj

doi:10.1016/j.crma.2007.07.011

Algebraic Geometry

Brauer obstruction for a universal vector bundle

Presented by: Gérard Laumon

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

Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 265-268.

Abstract
Résumé

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 $⋀^{r} E = ξ$ , where ξ is a fixed line bundle. We prove that the Brauer group $Br (M)$ is $Z / n Z$ , 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 $X \times M$ to a point of X.

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 $⋀^{r} E = ξ$ , où ξ est un fibré en droites donné. Nous montrons que le groupe de Brauer $Br (M)$ est égale à $Z / n Z$ , où $n = pgcd (r, \deg ξ)$ . 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 $X \times M$ en un point de X.

Received: 2006-08-17
Accepted: 2007-07-17
Published online: 2007-08-21

DOI: 10.1016/j.crma.2007.07.011

Author's affiliations:

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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     author = {Vikraman Balaji and Indranil Biswas and Ofer Gabber and Donihakkalu S. Nagaraj},
     title = {Brauer obstruction for a universal vector bundle},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {265--268},
     publisher = {Elsevier},
     volume = {345},
     number = {5},
     year = {2007},
     doi = {10.1016/j.crma.2007.07.011},
     language = {en},
}

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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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