# On the multicanonical systems of quasi-elliptic surfaces in characteristic 3

## TL;DR

This paper investigates the properties of multicanonical systems on quasi-elliptic surfaces in characteristic 3, establishing that for m ≥ 5, these systems define a quasi-elliptic fiber space, with 5 being the minimal such value.

## Contribution

It proves that for quasi-elliptic surfaces in characteristic 3, the multicanonical system |mK_S| induces a fiber space structure for all m ≥ 5, and 5 is the optimal bound.

## Key findings

- |mK_S| defines a fiber space for m ≥ 5
- The bound m=5 is optimal for such surfaces
- Characterization of multicanonical systems in characteristic 3

## Abstract

We consider the multicanonical systems $|mK_S|$ of quasi-elliptic surfaces with Kodaira dimension $1$ in characteristic 3. We show that for any $m \geq 5$ $|mK_S|$ gives the structure of quasi-elliptic fiber space, and the number $5$ is best possible to give the structure for any such surfaces.