On boundedness of indices of minimal pairs -- surfaces

TL;DR

This paper investigates the boundedness of indices of minimal pairs in algebraic geometry, showing that such indices cannot always be uniformly bounded solely by dimension and Cartier index, especially in the context of semi-ample canonical divisors.

Contribution

It demonstrates that, in general, the multiple of the canonical divisor cannot be expressed as a pullback of a Cartier divisor on the base, indicating limitations in boundedness results.

Findings

Boundedness of indices depends on more than just dimension and Cartier index.

Counterexamples show the impossibility of a uniform multiple n.

Highlights limitations in the theory of minimal pairs with semi-ample canonical divisors.

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 $Z$.