Suppose we have a one parameter family of smooth curves degenerating to a nodal curve, and a family of linear series on smooth fibres. By studying the 'limit' of the linear series on the singular fibre, we can obtain  information  about  linear  series  on smooth  fibres.  Here  we  need  a  suitable  notion  of  'limit'.

  In this talk, I will describe a 'weak' semistable degeneration of symmetric product of smooth curves: a space where linear series lie in. By taking closure of the linear series on smooth fibres in this space, we get a suitable 'limit'. We use Ziv Ran's results on relative Hilbert scheme of points for the above family of curves to get the 'weak' semistable degeneration. Here 'weak' means the central fibre may have components of multiplicity 2.