Ramified Covering Maps and Stability of Pulled-back Bundles

Abstract

Abstract Let $f\,:\,C\,\longrightarrow \,D$ be a nonconstant separable morphism between irreducible smooth projective curves defined over an algebraically closed field. We say that $f$ is genuinely ramified if ${\mathcal O}_D$ is the maximal semistable subbundle of $f_*{\mathcal O}_C$ (equivalently, the induced homomorphism $f_*\,:\, \pi _1^{\textrm{et}}(C)\,\longrightarrow \, \pi _1^{\textrm{et}}(D)$ of étale fundamental groups is surjective). We prove that the pullback $f^*E\,\longrightarrow \, C$ is stable for every stable vector bundle $E$ on $D$ if and only if $f$ is genuinely ramified.