We show that $|mK_X|$ defines a birational map and has no fixed part for some bounded positive integer $m$ for any $\frac{ 1}{ 2}$-lc surface $X$ such that $K_X$ is big and nef. For every positive integer $n\geq 3$, we construct a sequence of projective surfaces $X_{ n,i}$, such that $K_{ X_{ n,i}}$ is ample, ${ \rm{ mld}}(X_{ n,i})>\frac{ 1}{ n}$ for every $i$, $\lim_{ i\rightarrow+\infty}{ \rm{ mld}}(X_{ n,i})=\frac{ 1}{ n}$, and for any positive integer $m$, there exists $i$ such that $|mK_{ X_{ n,i}}|$ has non-zero fixed part. These results answer the surface case of a question of Xu.
Categories: math.AG