Fourier-Mukai transforms and normalisation of nodal curves

We study Arinkin's Poincar\'e sheaf $\mathcal{P}_C$ on the singular locus of $\overline{\mathsf{Jac}}_C$, the compactified Jacobian of rank one torsion-free sheaves on an integral nodal projective curve $C$. Each stratum of the singular locus $\mathsf{Sing}(\overline{\mathsf{Jac}}_C)$ is indexed by a partial normalisation $\Sigma \to C$. We prove that the Poincar\'e sheaf $\mathcal{P}_C$ restricted to each stratum can be expressed through the Poincar\'e sheaf $\mathcal{P}_{\Sigma}$, obtaining a relation between Fourier-Mukai transforms associated to $\mathcal{P}_C$ and $\mathcal{P}_{\Sigma}$. Our approach uses an intermediate geometry: the moduli space of parabolic modules of Bhosle and Cook, to intertwine sheaf data over the two curves. In a sequel, our formulae are used to study mirror symmetry in singular loci of Hitchin systems.