Polarized Pushouts for Finite Fields

Let p: Y–> X be a surjection between schemes projective over the algebraic closure of a finite field. Let L be a line bundle on X such that p^*(L) is globally generated. I give a natural necessary and sufficient condition under which some positive tensor power of L is globally generated. I apply this to give a sufficient condition for semi-ampleness of nef line bundles on the moduli space of stable pointed curves, in positive characteristic.