Poincaré and Picard bundles on the moduli spaces of bundles on curves (Q6607613)

Consider a smooth algenbraic curve and the moduli space of semistable bundles of prescribed rank and determinant. If the parameters are general (the rank and the degree of the determinant are coprime), one has an appripriately normalized universal bundle on the product of the curve and the moduli space, which is usually called the Poincaré bundle. Twist the Poincaré bundle by the pullback of a power of the ample line bundle assciated with a point on the curve, then consider the pushforward to the moduli space (there might be higher direct images, but they vanish for large enough twists). One obtains interesting sheaves (bundles in good cases) on the moduli space itself (opposed to the product). The paper of Bhosle gives a nice overview of various properties of Picard and Poincaré bundles both for smooth curves (questions studied by pleny of authors) and for nodal curves (most of these results obtained rather recently by the author). The overview starts with compactified Jacobians, where the Chern classes of Picard bundles are computed up to numerical equivalence in terms of (generalized) theta divisors. Next, ampleness and stability are studied and compared. Next, the author moves to the higher rank case and recalls results on stability and nefness of Poincaré and Picard bundles in the coprime case. In the non-coprime case the Poincaré bundle does not exist, but one can replace it with a projective version. Similar results are discussed for nodal curves. Finally, the paper ends with a generalization of some of the questions studied above to the case of parabolic bundles. Overall, the paper is a very good place to orient onself in the vast literature dedicated to the subject.\N\NFor the entire collection see [Zbl 1545.14003].