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 $m\geq N$ and $(X, \Delta)\in\mathfrak{D}$