Explicit Effective Birationality for Singular Surfaces

Abstract

We give some explicit upper bounds on the effective birationality of the canonical or anti-canonical system for a singular surface. In particular, we show that for any surface $X$ with $\epsilon$-lc singularity and the canonical divisor $K_X$ or the anti-canonical divisor $-K_X$ is big and nef, then there exists explicit natural numbers $m$ and $l$ depending only on $\epsilon$ such that $|m{K}_X|$ or $|-l{K}_X|$ defines a birational map. Although these explicit values are expected to be far from optimal, they are the first explicit upper bounds of this type for surfaces.

13 pages, comments are very welcome!