Explicit Effective Birationality for Singular Surfaces

TL;DR

This paper establishes explicit upper bounds on the natural numbers needed for the canonical or anti-canonical systems of singular surfaces to define birational maps, advancing understanding of effective birationality in algebraic geometry.

Contribution

It provides the first explicit upper bounds for effective birationality of singular surfaces with specific singularity and divisor conditions.

Findings

Explicit bounds depend only on the singularity parameter psilon.

Bounds apply to surfaces with big and nef canonical or anti-canonical divisors.

Results are the first of their kind for 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 $|mK_X|$ or $|-lK_X|$ 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.