The Picard group of the universal moduli space of vector bundles over the moduli space of stable curves.

Abstract

The thesis is divided in two chapters. In the rst one we construct the moduli stack of properly balanced vector bundles on semistable curves and we determine explicitly its Picard group. As a consequence, we obtain an explicit description of the Picard groups of the universal moduli stack of vector bundles on smooth curves and of the Schmitt's compacti cation over the stack of stable curves. We prove some results about the gerbe structure of the universal moduli stack over its rigidi cation by the natural action of the multiplicative group. In particular, we give necessary and su cient conditions for the existence of Poincar e bundles over the universal curve of an open substack of the rigidi cation, generalizing a result of Mestrano-Ramanan. In the second chapter we compute the Picard group of the universal abelian variety over the moduli stack Ag;n of principally polarized abelian varieties over C with a symplectic principal level n-structure. We then prove that over C the statement of the Franchetta conjecture holds in a suitable form for Ag,n.