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

Abstract: Let $f : X \rightarrow Y$ be a generically smooth nonconstant morphism between irreducible projective curves, defined over an algebraically closed field, which is \'etale 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: \begin{enumerate} \item The homomorphism of \'etale fundamental groups $$f_* : \pi_1^{\rm et}(X) \rightarrow\pi_1^{\rm et}(Y)$$ induced by $f$ is surjective. \item There is no nontrivial \'etale covering $\phi : Y' \rightarrow Y$ admitting a morphism $q: X \rightarrow Y'$ such that $\phi\circ q = f$. \item The fiber product $X\times_Y X$ is connected. \item $\dim H^0(X, f^f_ {\mathcal O}X)= 1$. \item ${\mathcal O}_Y \subset f*{\mathcal O}_X$ is the maximal semistable subsheaf. \item The pullback $f^*E$ of every stable sheaf $E$ on $Y$ is also stable. \end{enumerate}