Rational correspondences between moduli spaces of curves defined by Hurwitz spaces

By associating to a curve C and a the so-called trace curve and reduced trace curve we define two rational maps ϕ and from the Hurwitz space of admissible covers of genus g=2k and degree d=k+1 to moduli spaces and . We study the induced map of the divisor class group of and to the divisor class group of .