N\'eron models of $Pic^0$ via $Pic^0$

TL;DR

This paper introduces a novel approach to describing the Néron model of a Jacobian using twisted curves and Picard functors, providing new geometric and combinatorial insights into its structure.

Contribution

It presents a new description of Néron models via twisted curves and Picard functors, extending the understanding of their geometric and combinatorial properties.

Findings

Provides a new geometric interpretation of Néron model points.

Introduces a universal group scheme $Pic^{0,l}_g$ over a compactification of $M_g$.

Offers combinatorial descriptions of special fiber components.

Abstract

We provide a new description of the N\'eron model of the Jacobian of a smooth curve $C_K$ with stable reduction $C_R$ on a discrete valuation ring $R$ with field of fractions $K$. Instead of the regular semistable model, our approach uses the regular twisted model, a twisted curve in the sense of Abramovich and Vistoli whose Picard functor contains a larger separated subgroup than the usual Picard functor of $C_R$. In this way, after extracting a suitable $l$th root from the uniformizer of $R$, the pullback of the N\'eron model of the Jacobian represents a Picard functor $Pic^{0,l}$ of line bundles of degree zero on all irreducible components of a twisted curve. Over $R$, the group scheme $Pic^{0,l}$ descends to the N\'eron model yielding a new geometric interpretation of its points and new combinatorial interpretations of the connected components of its special fibre. Furthermore, by construction, $Pic^{0,l}$ is represented by a universal group scheme $Pic^{0,l}_{g}$ of line bundles of degree zero over a smooth compactification $\overline{M}_g^l$ of $M_g$ where all N\'eron models of smoothings of stable curves are cast together after base change.