Ramified covering maps of singular curves and stability of pulled back bundles

Let f : X ⟶ Y be a generically smooth nonconstant morphism between irreducible projective curves, defined over an algebraically closed field, which is étale on an open subset of Y that contains both the singular locus of Y and the image, in Y , of the singular locus of X . We prove that the following statements are equivalent: The homomorphism of étale fundamental groups f_* : π _1^et(X) ⟶ π _1^et(Y) induced by f is surjective. There is no nontrivial étale covering ϕ : Y' ⟶ Y admitting a morphism q: X ⟶ Y' such that ϕ∘ q = f . The fiber product X× _Y X is connected. H^0(X, f^*f_* 𝒪_X) = 1 . 𝒪_Y ⊂ f_*𝒪_X is the maximal semistable subsheaf. The pullback f^*E of every stable sheaf E on Y is also stable.