Year: 2024
Type: preprint
Abstract: We study Arinkin's Poincaré 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 $Σ\to C$. We prove that the Poincaré sheaf $\mathcal{P}_C$ restricted to each stratum can be expressed through the Poincaré sheaf $\mathcal{P}_Σ$, obtaining a relation between Fourier-Mukai transforms associated to $\mathcal{P}_C$ and $\mathcal{P}_Σ$. 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.
Source: arXiv (Cornell University)
Authors: Franco, Emilio, Hanson, Robert, Horn, Johannes, Oliveira, André
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4398192404