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Bulletin of the London Mathematical Society Advance Access originally published online on March 25, 2009
Bulletin of the London Mathematical Society 2009 41(3):458-472; doi:10.1112/blms/bdp017
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© 2009 London Mathematical Society

Stability of projective Poincaré and Picard bundles

I. Biswas

School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road
Bombay 400005
India
indranil@math.tifr.res.in

L. Brambila-Paz

CIMAT
Apdo. Postal 402
CP 36240
Guanajuato, Gto
Mexico
lebp@cimat.mx

P. E. Newstead

Department of Mathematical Sciences
The University of Liverpool
Peach Street
Liverpool
L69 7ZL
United Kingdom

Received 27 May 2008. Revision received 3 October 2008.

Let X be an irreducible smooth projective curve of genus g ≥ 3 defined over the complex numbers, and let M{xi} denote the moduli space of stable vector bundles on X of rank n and determinant {xi}, where {xi} is a fixed line bundle of degree d. If n and d have a common divisor, then there is no universal vector bundle on X x M{xi}. We prove that there is a projective bundle on X x M{xi} with the property that its restriction to X x {E} is isomorphic to P(E) for all E isin M{xi} and that this bundle (called the projective Poincaré bundle) is stable with respect to any polarization; moreover its restriction to {x} x M{xi} is also stable for any x isin X. We also prove stability results for bundles induced from the projective Poincaré bundle by homomorphisms PGL(n) -> H for any reductive H. We further show that there is a projective Picard bundle on a certain open subset M' of M{xi} for any d > n(g–1) and that this bundle is also stable. Also, we obtain new results on the stability of the Picard bundle even when n and d are coprime.

2000 Mathematics Subject Classification 14H60, 14J60.

All authors are members of the international research group VBAC. The second author acknowledges the support of CONACYT grant 48263-F.


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