| Let X be a smooth projective variety. Given any basepoint-free linear system, |D|, there is a dense open subset parametrizing smooth divisors, and over that subset, we can consider the relative Picard variety of the universal divisor, which parametrizes pairs of a smooth divisor in the linear system and a line bundle on that divisor. In the case where X is a surface, there is a natural compactification of the relative Picard variety, given by taking the moduli space of pure one-dimensional Gieseker-semistable sheaves with respect to some polarization. In the case of the projective plane, this is an irreducible projective variety of Picard number 2. We study the nef and effective cones of these moduli spaces, and talk about the relation with variation of Bridgeland stability conditions.;We show how the knowledge of the Picard group of this moduli space of pure one-dimensional sheaves on the projective plane can be used to deduce that every section of the relative Picard varieties of the complete linear system of plane curves (of degree at least 3) comes from restriction a line bundle on the plane. We also give an independent proof of this fact for any basepoint-free linear system on any smooth projective variety when the locus of reducible or non-reduced divisors in the linear system has codimension at least two. |
