On boundedness of indices of minimal pairs -- surfaces

Abstract

For given positive integers $d$ and $m$, consider the projective klt pairs $(X,B)$ of dimension $d$, of Cartier index $m$, and with semi-ample $K_X+B$ defining a contraction $\pi\colon X\to Z$. We prove that it is not possible in general to write $n(K_X+B)\sim\pi^*A_Z$ for some $n$ depending only on $d$ and $m$, and some Cartier divisor $A_Z$ on $Z$.