Mathematical Research Letters

Volume 18 (2011)

Number 4

Sequences of LCT-Polytopes

Pages: 733 – 746



Anatoly Libgober (Department of Mathematics, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607-7045, USA)

Mircea Mustaţă (Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA)


To $r$ ideals on a germ of smooth variety $X$ one attaches a rational polytope in $\RR_+^r$ (the \emph{LCT-polytope}) that generalizes the notion of log canonical threshold in the case of one ideal. We study these polytopes, and prove a strong form of the Ascending Chain Condition in this setting: we show that if a sequence $(P_m)_{m\geq 1}$ of LCT-polytopes in $\RR_+^r$ converges to a compact subset $Q$ in the Hausdorff metric, then $Q=\bigcap_{m\geq m_0}P_m$ for some $m_0$, and $Q$ is an LCT-polytope.

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