Genuinely ramified maps and monodromy

For any genuinely ramified morphism f : Y --> X between irreducible smooth projective curves we prove that (Y xX Y ) \ Delta is connected, where Delta C Y xX Y is the diagonal. Using this result the following are proved: (1) If f is further Morse then the Galois closure is the symmetric group Sd, where d = degree(f). (2) The Galois group of the general projection, to a line, of any smooth curve X C Pn of degree d, which is not contained in a hyperplane and contains a non-flex point, is Sd.