Graded families of ideals and convex regions

Haoxi Hu

Graded families of ideals and convex regions

Commutative Algebra 2026-07-06 v1 Algebraic Geometry Combinatorics

Authors: Haoxi Hu

Abstract

We study the interplay between graded families of ideals in $\mathbb{K}$ -domains and their associated convex regions. These regions, called Newton-Okounkov regions, arise naturally from graded families of ideals associated to a valuation with one-dimensional leaves. Our main focus is to compute asymptotic resurgence number of a pair of graded families of ideals. By combining techniques from Attouch--Wets topology and convex-geometric properties of Newton-Okounkov regions, we characterize the asymptotic resurgence number through containment relations between the pair of corresponding Newton-Okounkov regions.

Keywords

graded families of ideals Newton-Okounkov regions asymptotic resurgence valuations convex geometry Attouch-Wets topology

Cite

@article{arxiv.2607.05556,
  title  = {Graded families of ideals and convex regions},
  author = {Haoxi Hu},
  journal= {arXiv preprint arXiv:2607.05556},
  year   = {2026}
}