Fourier-Mukai transforms and normalisation of nodal curves

We study Arinkin’s Poincaré sheaf ${\mathcal{P}}_C$ on the singular locus of ${\overline{\mathsf{J}\mathsf{a}\mathsf{c}}}_C$, the compactified Jacobian of rank one torsion-free sheaves on an integral nodal projective curve $C$. Each stratum of the singular locus $\mathsf{S}\mathsf{i}\mathsf{n}\mathsf{g}({\overline{\mathsf{J}\mathsf{a}\mathsf{c}}}_C)$ is indexed by a partial normalisation $\mathrm{\Sigma }\to C$. We prove that the Poincaré sheaf ${\mathcal{P}}_C$ restricted to each stratum can be expressed through the Poincaré sheaf ${\mathcal{P}}_{\mathrm{\Sigma }}$, obtaining a relation between Fourier-Mukai transforms associated to ${\mathcal{P}}_C$ and ${\mathcal{P}}_{\mathrm{\Sigma }}$. 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.