Fourier-Mukai and autoduality for compactified Jacobians, II

摘要

To every reduced (projective) curve X with planar singularities one can associate, following E Esteves, many fine compactified Jacobians, depending on the choice of a polarization on X, which are birational (possibly nonisomorphic) Calabi-Yau projective varieties with locally complete intersection singularities. We define a Poincare sheaf on the product of any two (possibly equal) fine compactified Jacobians of X and show that the integral transform with kernel the Poincare sheaf is an equivalence of their derived categories, hence it defines a Fourier-Mukai transform. As a corollary of this result, we prove that there is a natural equivariant open embedding of the connected component of the scheme parametrizing rank-1 torsion-free sheaves on X into the connected component of the algebraic space parametrizing rank-1 torsion-free sheaves on a given fine compactified Jacobian of X.