On the fixed part of pluricanonical systems for surfaces

We show that |mKX|$|mK_X|$ defines a birational map and has no fixed part for some bounded positive integer m for any 12$\frac{1}{2}$‐lc surface X such that KX$K_X$ is big and nef. For every positive integer n≥3$n\ge 3$, we construct a sequence of projective surfaces Xn,i$X_{n,i}$, such that KXn,i$K_{X_{n,i}}$ is ample, mld(Xn,i)>1n${\rm {mld}}(X_{n,i})>\frac{1}{n}$ for every i, limi→+∞mld(Xn,i)=1n$\lim _{i\rightarrow +\infty }{\rm {mld}}(X_{n,i})=\frac{1}{n}$, and for any positive integer m, there exists i such that |mKXn,i|$|mK_{X_{n,i}}|$ has nonzero fixed part. These results answer the surface case of a question of Xu.