Rational correspondences between moduli spaces of curves defined by   Hurwitz spaces
  By associating to a curve C of genus g=2k and a pencil of degree d=k+1 the so-called trace curve (resp. the reduced trace curve) we define a rational map from the Hurwitz space of admissible covers of genus g=2k and degree d=k+1 to a moduli space of stable curves. We study the induced map between the divisor class groups of these moduli spaces of curves. 
Gerard van der Geer and Alexis Kouvidakis
math.AG