Pushforward of structure sheaf and virtual global generation

Abstract

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

Final version