The faithfulness of the monodromy representations associated with certain families of algebraic curves

We consider the faithfulness of the monodromy representation associated with the universal family of n-pointed algebraic curves of genus g (2−2g−n<0). We shall show that the restriction ρg,n of the representation to the geometric fundamental group is faithful for g=0,1. We shall also show that if ρg,n is faithful, then so is ρg,n+1.