Roots of line bundles on curves and Néron d-models

Over an algebraic field K with discrete valuation, we study the line bundles on a smooth curve CK whose rth power is isomorphic to a given line bundle FK . For FK = O, these objects form a finite group K-scheme O/r; furthermore, as soon as the degree of FK is a multiple of r, we get a finite K-torsor FK/r under the group scheme O/r. On the discrete valuation ring R ⊂ K, there exist Néron models N(O/r) and N(FK/r), which are universal R-models of O/r and FK/r in the sense of the Néron mapping property. In general, N(O/r) has a group structure, but the properness may be lost; furthermore, N(FK/r) is in general not proper and not a torsor. In the present work, we cast the notion of Néron models on a stack theoretic base S[d]: a proper cover of S = SpecR, invertible on K, and with stabilizer of order d on the closed point. For all d, we show that there exist Néron d-models Nd(O/r) andNd(FK/r) on S[d] which are universal S[d]-models in the sense of the Néron mapping property. Assume that O/r is tamely ramified on R; then, for a suitable d, Nd(O/r) is a finite group stack on S[d]. Furthermore, d can be chosen so that Nd(O/r) represents the r-th roots of O on a twisted curve C → S[d] extending CK (a kind of stack-theoretic curve introduced by Abramovich and Vistoli). Similarly, given a line bundle FK on CK with FK/r tamely ramified on R, for a suitable d we get a finite torsor Nd(FK/r) under Nd(O/r). Under suitable conditions on d, Nd(FK/r) represents F/r, the finite torsor of rth roots of a line bundle on a twisted curve F → C extending FK → CK from K to S[d]. We treat the problem of quantifying and minimizing the choice of d.