Pushforward of structure sheaf and virtual global generation

Let f\,:\,X\, \,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_* OX)/ OY)^* is virtually globally generated. Moreover, ((f_* OX)/ OY)^* is ample if and only if f is genuinely ramified.