# Relative semi-ampleness in positive characteristic

## TL;DR

This paper proves that in positive characteristic, fibrewise semi-ampleness of an invertible sheaf implies relative semi-ampleness, highlighting a key difference from characteristic zero.

## Contribution

It establishes a new semi-ampleness criterion in positive characteristic that does not hold in characteristic zero.

## Key findings

- Fibrewise semi-ampleness implies relative semi-ampleness in positive characteristic.
- The statement fails in characteristic zero.
- Highlights differences between positive characteristic and zero in algebraic geometry.

## Abstract

Given an invertible sheaf on a fibre space between projective varieties of positive characteristic, we show that fibrewise semi-ampleness implies relative semi-ampleness. The same statement fails in characteristic zero.