标题： 显式有效双有理性对于奇异曲面 显示英文标题

标题： Explicit Effective Birationality for Singular Surfaces

摘要： 我们给出关于奇异曲面的典范或反典范系统有效双有理性的某些显式上界。 特别是，我们证明对于任何具有$\epsilon$-lc奇点的曲面$X$，如果其典范除子$K_X$或反典范除子$-K_X$是大且 nef 的，则存在仅依赖于$\epsilon$的显式自然数$m$和$l$，使得$|mK_X|$或$|-lK_X|$定义一个双有理映射。 尽管这些显式值预计远非最优，但它们是针对曲面的此类显式上界的第一批结果。 显示英文摘要

摘要： 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.