Stability of Projective Poincaré and Picard Bundles

Let X be an irreducible smooth projective curve of genus g ≥ 3 defined over the complex numbers and let Mξ 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, there is no universal vector bundle on X ×Mξ. We prove that there is a projective bundle on X ×Mξ with the property that its restriction to X × {E} is isomorphic to P (E) for all E ∈ Mξ and that this bundle (called the projective Poincaré bundle) is stable with respect to any polarization; moreover its restriction to {x} ×Mξ is also stable for any x ∈ X . We prove also stability results for bundles induced from the projective Poincaré bundle by homomorphisms PGL(n) → H for any reductive H . We show further that there is a projective Picard bundle on a certain open subset M of Mξ for any d > n(g − 1) and that this bundle is also stable. We obtain new results on the stability of the Picard bundle even when n and d are coprime.