Sequences of LCT-polytopes

TL;DR

This paper investigates sequences of LCT-polytopes associated with ideals on smooth varieties, proving a strong ascending chain condition that ensures convergence properties and the stability of these polytopes.

Contribution

It establishes a strong form of the Ascending Chain Condition for sequences of LCT-polytopes, showing their limits are also LCT-polytopes and characterizing their convergence behavior.

Findings

Sequences of LCT-polytopes converge to an LCT-polytope under Hausdorff metric.

Limit sets of converging sequences are intersections of the tail of the sequence.

The paper proves a strong Ascending Chain Condition for these polytopes.

Abstract

To r ideals on a germ of smooth variety X one attaches a rational polytope in the r-dimensional Euclidean space (the 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 of such LCT-polytopes converges to a compact subset Q in the Hausdorff metric, then Q is equal to the intersection of all but finitely many of the P_m. Furthermore, Q is an LCT-polytope.