2015/09/22 by Alessandro Chiodo, Chiodo, Alessandro
Mathematics · #14H10 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14H10
paper · pdf · doi:10.48550/arxiv.1509.06483
23 pages
arxiv created 2015/09/22 · arxiv updated 2015/09/23
We provide a new description of the Néron model of the Jacobian of a smooth curve CK with stable reduction CR 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 CR. In this way, after extracting a suitable lth root from the uniformizer of R, the pullback of the Néron model of the Jacobian represents a Picard functor Pic0,l of line bundles of degree zero on all irreducible components of a twisted curve. Over R, the group scheme Pic0,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, Pic0,l is represented by a universal group scheme Pic0,lg of line bundles of degree zero over a smooth compactification Mgl of Mg where all Néron models of smoothings of stable curves are cast together after base change.