Academic literature on the topic 'Relative compactified Jacobians'

We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.

Let Y → ℙn be a flat family of reduced Gorenstein curves, such that the compactified relative Jacobian [Formula: see text] is a Lagrangian fibration. We prove that X is a Beauville–Mukai integrable system if n = 3, 4, or 5, and the curves are irreducible and non-hyperelliptic. We also prove that X is a Beauville–Mukai system if n = 3, d is odd, and the curves are canonically positive 2-connected hyperelliptic curves.

The thesis presents a construction of singular holomorphically symplectic varieties as Lagrangian fibrations. They are relative compactified Prym varieties associated to curves on symplectic surfaces with an antisymplectic involution. They are identified with the fixed locus of a symplectic involution on singular moduli spaces of sheaves of dimension 1. An explicit example, giving a singular irreducible symplectic 6-fold without symplectic resolutions, is described for a K3 surface which is the double cover of a cubic surface. In the case of abelian surfaces, a variation of this construction is studied to get irreducible symplectic varieties: relative compactified 0-Prym varieties. A partial classification result is obtained for involutions without fixed points: either the 0-Prym variety is birational to an irreducible symplectic variety of K3[n]-type, or it does not admit symplectic resolutions.