Picard Groups of the Moduli Spaces of Vector Bundles over Algebraic Surfaces

The purpose of this note is to determine the Picard group of the moduli space of vector bundles over an arbitrary algebraic surface. Since Donaldson’s pioneer work on using moduli of vector bundles to define smooth invariants of an algebraic surface, there has been a surge of interest in understanding the geometry of this moduli space. Among other things, the study of line bundles on this moduli space plays a major role in this area. One important question remain open until now is to determine the Picard group of this moduli space. This is known in some special cases, for instance for projective plane [St], ruled surfaces [Qi2, Yo] and K3 surfaces [GH]. In this note, we will settle this question by providing a general construction of line bundles that will include virtually all line bundles on this moduli space, when the second Chern class of the sheaves parameterized by this moduli space is sufficiently large. The construction is again based on Knudsen and Mumford’s recipe of determinant line bundle construction. The new input from this note is that instead of using complexes on surface X we will use complexes on X×X, which yields all previously known line bundles as well as new ones. After this construction, we will use our knowledge of the first two Betti numbers of this moduli space to argue that this construction contains virtually all line bundles on this moduli space. The proof relies heavily on the Grothendieck-Riemann-Roch theorem and the knowledge of the singularities of the moduli space.