A pathology of asymptotic multiplicity in the relative setting

TL;DR

The paper presents an example demonstrating that the asymptotic multiplicity can be infinite in the relative setting, indicating limitations in the divisorial Zariski decomposition for pseudoeffective divisors.

Contribution

It provides a specific example showing the failure of the divisorial Zariski decomposition in the relative setting due to infinite asymptotic multiplicity.

Findings

Example of infinite asymptotic multiplicity in a relative setting

Shows divisorial Zariski decomposition may not be defined in this context

Highlights limitations in current divisor decomposition theories

Abstract

We point out an example of a projective family $\pi : X \to S$, a $\pi$-pseudoeffective divisor $D$ on $X$, and a subvariety $V \subset X$ for which the asymptotic multiplicity $\sigma_V(D;X/S)$ is infinite. This shows that the divisorial Zariski decomposition is not always defined for pseudoeffective divisors in the relative setting.