Néron models of $Pi{c}^0$ via $Pi{c}^0$

Abstract

We provide a new description of the Néron 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 Néron model of the Jacobian represents a Picard functor $Pi{c}^{0,l}$ of line bundles of degree zero on all irreducible components of a twisted curve. Over $R$, the group scheme $Pi{c}^{0,l}$ descends to the Néron model yielding a new geometric interpretation of its points and new combinatorial interpretations of the connected components of its special fibre. Furthermore, by construction, $Pi{c}^{0,l}$ is represented by a universal group scheme $Pi{c}_g^{0,l}$ of line bundles of degree zero over a smooth compactification ${\overline{M}}_g^l$ of $M_g$ where all Néron models of smoothings of stable curves are cast together after base change.

23 pages

References in corpus (1)

• Moduli of Generalized Line Bundles on a Ribbon

Cited by in corpus (3)

• Compactifications of the universal Jacobian over curves with marked points
• Gluing semi-orthogonal decompositions
• Universal Néron models for curves with marked points