A new class of compactified Jacobians for families of reduced curves

TL;DR

This paper introduces a new framework for constructing compactified Jacobians for families of reduced curves, extending classical methods and ensuring scheme structures under mild conditions.

Contribution

It provides an abstract definition of relative compactified Jacobians, introduces V-stability conditions, and extends classical compactification techniques to broader curve families.

Findings

Fibers of the relative Jacobian are schemes under mild assumptions.

V-stability conditions yield well-behaved compactified Jacobians.

For curves with planar singularities, the new compactifications share classical properties.

Abstract

This is the first paper of a series of three. Here we give an abstract definition of the relative compactified Jacobian of a family of reduced curves. We prove that, under some mild assumptions on the family of curves, the fibres of the relative Jacobian are schemes (and not just algebraic spaces). We define V-stability conditions, and use them to construct relative compactified Jacobians. This extends the classical methods to produce modular compactifications of the Jacobian. To conclude, we show that, in the case when the curves have at worst planar singularities, the compactified Jacobians constructed from V-stability conditions have the same good properties of the classical ones.