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

Abstract

Let $f:X\to Y$ 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: \begin{enumerate} \item The homomorphism of étale fundamental groups $$f_{\ast }:\ddot{I}{€}_1^{\mathrm{et}}(X)\to \ddot{I}{€}_1^{\mathrm{et}}(Y)$$ induced by $f$ is surjective. \item There is no nontrivial étale covering $\ddot{I}•:Y^{\prime }\to Y$ admitting a morphism $q:X\to Y^{\prime }$ such that $\ddot{I}•\circ q=f$. \item The fiber product $X{\times }_{Y}X$ is connected. \item $\dim H^0(X,f^{\ast }{f}_{\ast }{\mathcal{O}}_X)=1$. \item ${\mathcal{O}}_Y\subset f_{\ast }{\mathcal{O}}_X$ is the maximal semistable subsheaf. \item The pullback $f^{\ast }E$ of every stable sheaf $E$ on $Y$ is also stable. \end{enumerate}

Final version