Boundedness of log-pluricanonical maps for surfaces of log-general types in positive characteristic

Abstract

In this article we prove the following boundedness result: Fix a DCC set $I\subset [0,1]$. Let $\mathfrak{D}$ be the set of all log pairs $(X,\Delta )$ satisfying the following properties: (i) $X$ is a projective surface defined over an algebraically closed field, (ii) $(X,\Delta )$ is log canonical and the coefficients of $\Delta$ are in $I$, and (iii) $K_X+\Delta$ is big. Then there is a positive integer $N=N(I)$ depending only on the set $I$ such that the linear system $|\lceil m(K_X+\Delta )\rceil |$ defines a birational map onto its image for all $m\ge N$ and $(X,\Delta )\in \mathfrak{D}$

Minor changes. Final version. To appear in the Osaka Journal of Mathematics