Fine compactified Jacobians of reduced curves

TL;DR

This paper explores the geometric properties of fine compactified Jacobians of reduced curves, especially those with locally planar singularities, and examines their universal forms and twisted Abel maps.

Contribution

It provides new insights into the structure and diversity of fine compactified Jacobians for singular curves, including examples of non-isomorphic cases and their universal counterparts.

Findings

Examples of non-isomorphic fine compactified Jacobians for nodal curves.

Analysis of universal fine compactified Jacobians over deformation spaces.

Investigation of twisted Abel maps into these Jacobians.

Abstract

To every singular reduced projective curve X one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, each of which yields a modular compactification of a disjoint union of the generalized Jacobian of X. We investigate the geometric properties of fine compactified Jacobians focusing on curves having locally planar singularities. We give examples of nodal curves admitting non isomorphic (and even non homeomorphic over the field of complex numbers) fine compactified Jacobians. We study universal fine compactified Jacobians, which are relative fine compactified Jacobians over the semiuniversal deformation space of the curve X. Finally, we investigate the existence of twisted Abel maps with values in suitable fine compactified Jacobians.