  In this paper, we study the Picard group of the moduli space of semistable
sheaves on a smooth quadric surface. We polarize the surface by an ample
divisor close to the anticanonical class. We focus especially on moduli spaces
of sheaves of small discriminant, where we observe new and interesting
behavior. Our method relies on constructing certain resolutions for semistable
sheaves and applying techniques of geometric invariant theory to the resulting
families of sheaves.
