Families of n-gonal curves with maximal variation of moduli
  We study families of n-gonal curves with maximal variation of moduli, which have a rational section. Certain numerical results on the degree of the modular map are obtained for such families of hyperelliptic and trigonal curves. In the last case we use the description of the relative Picard group of the universal family of trigonal curves. 
Sergey Gorchinskiy, Filippo Viviani