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

Abstract

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