Fine compactified Jacobians

TL;DR

This paper investigates fine compactified Jacobians of nodal curves, establishing their relation to Néron models, describing their stratification, and comparing them with coarse compactifications, thus advancing the understanding of their modular properties.

Contribution

It provides a proof linking the smooth locus of fine compactified Jacobians to Néron models and describes their stratification and quotient structures, offering new insights into their modular compactifications.

Findings

The smooth locus of a fine compactified Jacobian is isomorphic to the Néron model.

Fine compactified Jacobians admit stratifications via partial normalizations.

They can be realized as quotients of smooth loci of blowup Jacobians.

Abstract

We study Esteves's fine compactified Jacobians for nodal curves. We give a proof of the fact that, for a one-parameter regular local smoothing of a nodal curve $X$, the relative smooth locus of a relative fine compactified Jacobian is isomorphic to the N\'eron model of the Jacobian of the general fiber, and thus it provides a modular compactification of it. We show that each fine compactified Jacobian of $X$ admits a stratification in terms of certain fine compactified Jacobians of partial normalizations of $X$ and, moreover, that it can be realized as a quotient of the smooth locus of a suitable fine compactified Jacobian of the total blowup of $X$. Finally, we determine when a fine compactified Jacobian is isomorphic to the corresponding Oda-Seshadri's coarse compactified Jacobian.