Finite generation of the $\log$ canonical ring for $3$-folds in char $p$

We prove that the $\log$ canonical ring of a projective klt pair of dimension $3$ with $\mathbb{Q}$-boundary over an algebraically closed field of characteristic $p \gt 5$ is finitely generated. In the process we prove $\log$ abundance for such pairs in the case $\kappa = 2$.

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Received 9 June 2015

Accepted 28 March 2016

Published 1 September 2017