Algebraic Geometry Seminar

Over a family of varieties in which some fibers are singular, the relative Picard stack (the moduli space of line bundles) may fail to be compact. I'll discuss an asymptotic stability condition aimed at compactifying it, which generalizes the GIT stability condition used in the 1990s by Caporaso in compactifying relative Picard schemes over families of curves. I'll give some results on counting semistable line bundles on reducible varieties of arbitrary dimension with ample or anti-ample canonical bundle, as well as similar results over degenerate K3 surfaces of Type II. I'll also discuss some open questions related to constructing proper moduli spaces using semistable line bundles.