Effective birationality for sub-pairs with real coefficients

Abstract

For $\hat{I}\mu$-lc Fano type varieties $X$ of dimension $d$ and a given finite set $\hat{I}“$, we show that there exists a positive integer $m_0$ which only depends on $\hat{I}\mu ,d$ and $\hat{I}“$, such that both $|-m{K}_X-{\sum }_i\lceil m{b}_i\rceil B_i|$ and $|-m{K}_X-{\sum }_i\lfloor m{b}_i\rfloor B_i|$ define birational maps for any $m\ge m_0$ provided that $B_i$ are pseudo-effective Weil divisors, $b_i\in \hat{I}“$, and $-(K_X+{\sum }_i{b}_i{B}_i)$ is big. When $\hat{I}“\subset [0,1]$ satisfies the DCC but is not finite, we construct an example to show that the effective birationality may fail even if $X$ is fixed, $B_i$ are fixed prime divisors, and $(X,B)$ is $\hat{I}{\mu }^{\prime }$-lc for some $\hat{I}{\mu }^{\prime }>0$.

21 pages, comments are welcome!