The relative Fujita-Zariski theorem

We prove, with no claim to originality, a relative version of the Fujita-Zariski theorem. When the base is a field, this result is due to Fujita (1983) and states that if an invertible sheaf on a proper variety is ample on its base locus, its sufficiently high powers are globally generated. The special case where the base locus is finite was proved by Zariski (1962), whence the name.