Sequences of LCT-Polytopes

Abstract

To r ideals on a germ of smooth variety X one attaches a rational polytope in R r + (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 ) m≥1 of LCT-polytopes in R r + converges to a compact subset Q in the Hausdorff metric, then Q = m≥m0 P m for some m 0 , and Q is an LCT-polytope.

ACC for log canonical thresholds

“…Under the above notation, we have the following important result on log canonical thresholds. Log canonical threshold can be generalized to the case of multiple divisors (called testing divisors) and we get LCT-polytopes (see [LM11]).…”

Constancy regions of mixed multiplier ideals in two‐dimensional local rings with rational singularities

“…, 0). The above result for this case gives a mild generalization of the well-known formula for the region R a a a (λ λ λ 0 ) in the smooth case (see [14] where this region is denoted LCT-polytope). Namely, it is the rational convex polytope determined by the inequalities e 1,j z 1 + • • • + e r,j z r < k j + 1 + e λ λ λ 0 j , corresponding to either rupture or dicritical divisors E j .…”

“…Remark When X has a rational singularity at O , we may have a strict inclusion for . The above result for this case gives a mild generalization of the well‐known formula for the region in the smooth case (see where this region is denoted LCT‐polytope). Namely, it is the rational convex polytope determined by the inequalities corresponding to either rupture or dicritical divisors .…”

“…The case of mixed multiplier ideals has not received as much attention as multiplier ideals. Libgober and Mustaţǎ studied properties of the jumping wall associated with the constancy region of the origin . Naie in uses mixed multiplier ideals in order to study the irregularity of abelian coverings of smooth projective surfaces.…”

Multivariable Hodge theoretical invariants of germs of plane curves. II

“…In [18], it is proven that the Ascending Chain Condition holds for LCT-polytopes, that is that all increasing chain of LCT-polytopes is eventually stationary. We give an example to show that it is no more true for the polytopes of quasi-adjunction associated to a φ = 1.…”