On the multicanonical systems of quasi-elliptic surfaces

We consider the multicanonical systems $\mathrm{\mid }m{K}_S\mathrm{\mid }$ of quasi-elliptic surfaces with Kodaira dimension $1$ in characteristic 2. We show that for any $m\ge 6$ $\mathrm{\mid }m{K}_S\mathrm{\mid }$ gives the structure of quasi-elliptic fiber space, and 6 is the best possible number to give the structure for any such surfaces.