Pushforward of structure sheaf and virtual global generation

Abstract: Let $f:X\rightarrow Y$ be a generically smooth morphism between irreducible smooth projective curves over an algebraically closed field of arbitrary characteristic. We prove that the vector bundle $((f_{\mathcal O}_X)/{\mathcal O}_Y)^$ is virtually globally generated. Moreover, $((f_{\mathcal O}_X)/{\mathcal O}_Y)^$ is ample if and only if $f$ is genuinely ramified.