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Let  X  be an irreducible smooth projective curve of genus  g  ⩾ 3 defined over the complex numbers, and let ℳ ξ  denote the moduli space of stable vector bundles on  X  of rank  n  and determinant ξ, where ξ 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  × ℳ ξ . We prove that there is a projective bundle on  X  × ℳ ξ  with the property that its restriction to  X  × { E } is isomorphic to  P ( E ) for all  E  ∈ ℳ ξ  and that this bundle (called the projective Poincaré bundle) is stable with respect to any polarization; moreover its restriction to { x } × ℳ ξ  is also stable for any  x  ∈  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 ξ  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.

Stability of projective Poincaré and Picard bundles