Ramified covering maps and stability of pulled back bundles
Let $f:C\rightarrow D$ 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 homomorphism of etale fundamental groups is surjective). We prove that the pullback $f^*E\rightarrow C$ is stable for every stable vector bundle $E$ on $D$ if and only if $f$ is genuinely ramified.
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Algebraic Geometry
14H30, 14H60, 14E20
