  A family of curves over a discrete valuation ring is called semi-factorial if
every line bundle on the generic fibre extends to a line bundle on the total
space. In the nodal case, 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 centered at the
non-regular closed points yields a semi-factorial model of the generic fibre.
  As an application, we extend Raynaud's construction of the N\'eron
(lft)-model of the jacobian of the generic fibre of a family of nodal curves to
the case where the generic fibre is singular.
