On the fixed part of pluricanonical systems for surfaces

Abstract

We show that $|m{K}_X|$ 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 $n\ge 3$, we construct a sequence of projective surfaces $X_{n,i}$, such that $K_{X_{n,i}}$ is ample, $\mathrm{mld}(X_{n,i})>\frac{1}{n}$ for every $i$, ${\lim }_{i\to +\infty }\mathrm{mld}(X_{n,i})=\frac{1}{n}$, and for any positive integer $m$, there exists $i$ such that $|m{K}_{X_{n,i}}|$ has non-zero fixed part. These results answer the surface case of a question of Xu.

20 pages, v1