Genuinely ramified maps and monodromy

For any genuinely ramified morphism $f:Y⟶X$ between irreducible smooth projective curves we prove that $\overline{(Y{\times }_{X}Y)\setminus \mathrm{\Delta }}$ is connected, where $\mathrm{\Delta }\subset Y{\times }_{X}Y$ is the diagonal. Using this result the following are proved: If $f$ is further Morse then the Galois closure is the symmetric group $S_d$, where $d=\text{degree}(f)$. The Galois group of the general projection, to a line, of any smooth curve $X\subset {\text{PP}}^n$ of degree $d$, which is not contained in a hyperplane and contains a non-flex point, is $S_d$.