Genuinely ramified maps and monodromy

Abstract

For any genuinely ramified morphism f:Y⟶X between irreducible smooth projective curves we prove that (Y×XY)∖Δ‾ is connected, where Δ⊂Y×XY 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⊂Pn of degree d, which is not contained in a hyperplane and contains a non-flex point, is Sd.