On the fixed part of pluricanonical systems for surfaces

TL;DR

This paper proves that for certain surfaces, pluricanonical systems define birational maps without fixed parts for some bounded m, and constructs examples showing fixed parts can appear for arbitrarily large m, answering a question of Xu.

Contribution

It establishes boundedness of fixed parts in pluricanonical systems for lc surfaces and constructs examples with fixed parts for large m, addressing a key open question.

Findings

$|mK_X|$ is birational and fixed part free for some bounded m on certain surfaces.

Constructs sequences of surfaces with fixed parts appearing in pluricanonical systems for arbitrarily large m.

Answers the surface case of Xu's question regarding fixed parts in pluricanonical systems.

Abstract

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.