Semi-factorial nodal curves and Néron models of jacobians

Abstract

Abstract. Following Pépin, we call a family of curves over a discrete valuation ring semifactorial if every line bundle on the generic fibre extends to a line bundle on the total space. In the case of nodal curves with split singularities, we give a sufficient and necessary condition for semi-factoriality, in terms of combinatorics of the dual graph of the special fibre. In particular, we show that performing one blow-up with center the non-regular closed points yields a semi-factorial model of the generic fibre. As an application, we extend the result of Raynaud relating Néron models of smooth curves and Picard functors of their regular models to the case of (possibly singular) curves having a semi-factorial model.

On Jacobians of geometrically reduced curves and their Néron models

“…Hence we shall now investigate under which circumstances the S-scheme P sep C already is the Néron lft-model of Pic 0 C/K . For nodal curves, a similar question was studied by Orecchia [20]. We also investigate the existence of closely related semi-factorial models introduced by Pépin [ for closed points x 1 , ..., x n of C .…”

“…Let R be a discrete valuation ring with field of fractions K and let C → S := Spec R be a proper and flat morphism whose fibres are nodal curves with split singularities ([20], Definitions 1.1 and 1.2). Orecchia [20] studied the question when C is a Néron-Picard model of its generic fibre. Basically, his result (as stated in [20]) can be paraphrased as follows: We let Γ be the dual graph of the special fibre of C → S. We consider the labelled graph (Γ, l) of C → S, where each edge of Γ is labelled by the thickness of the corresponding singularity of the special fibre (see [20], Definition 6.1).…”

“…Orecchia [20] studied the question when C is a Néron-Picard model of its generic fibre. Basically, his result (as stated in [20]) can be paraphrased as follows: We let Γ be the dual graph of the special fibre of C → S. We consider the labelled graph (Γ, l) of C → S, where each edge of Γ is labelled by the thickness of the corresponding singularity of the special fibre (see [20], Definition 6.1). The thickness measures how singular a singularity of the special fibre is when considered as a point of C .…”

“…This is an element of N ∪{∞} which is equal to 1 if and only if the corresponding point of C is regular, and equal to ∞ if and only if the corresponding point on C is a specialisation of a node on the generic fibre of C . We call (Γ, l) circuit-coprime if the labels appearing in any circuit in the resulting graph have no common prime divisor (see the Definition in [21] which replaces [20], Definition 5.19). Now Theorem 7.6 of [20] states that C → S is a Néron-Picard model of its generic fibre if and only if (Γ, l) is circuit-coprime (this is true as originally stated with the new definition of curcuit-coprimality from [21]).…”

“…We call (Γ, l) circuit-coprime if the labels appearing in any circuit in the resulting graph have no common prime divisor (see the Definition in [21] which replaces [20], Definition 5.19). Now Theorem 7.6 of [20] states that C → S is a Néron-Picard model of its generic fibre if and only if (Γ, l) is circuit-coprime (this is true as originally stated with the new definition of curcuit-coprimality from [21]). The example from above illustrates this result: Indeed, let C be the model of C from the Example above (constructed in the first push-out diagram).…”