Mathematical Research Letters

Volume 23 (2016)

Number 5

A pathology of asymptotic multiplicity in the relative setting

Pages: 1433 – 1451



John Lesieutre (Institute for Advanced Study, Princeton, New Jersey, U.S.A.; and University of Illinois, Chicago. Il., U.S.A.)


We point out an example of a projective family $\pi : X \to S$, a subvariety $V \subset X$, and a $\pi$-pseudoeffective divisor $D$ on $X$ 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.

Accepted 22 October 2015

Published 12 January 2017