Overview
This study focuses on the interaction between graded families of ideals, defined within $\mathbb{K}$-domains, and their corresponding Newton-Okounkov regions. These regions are convex and emerge from graded families of ideals that are linked to a valuation possessing one-dimensional leaves. The central objective of the research is to compute the asymptotic resurgence number for a specified pair of graded families of ideals. The methodology integrates concepts from Attouch-Wets topology with the convex-geometric attributes of Newton-Okounkov regions to characterize this number.
Research Context
The research situates itself within the algebraic structures of graded families of ideals in $\mathbb{K}$-domains. These algebraic constructs are recognized for having associated Newton-Okounkov regions, which are inherently convex. The connection between the graded families of ideals and these convex regions is established through their association with a valuation that features one-dimensional leaves. The concept of an asymptotic resurgence number is a key focus, specifically concerning its calculation for pairs of graded families of ideals. This implies an investigation into how properties of these families manifest in numerical invariants.
Approach
The approach to addressing the problem involves a multi-faceted strategy. Primarily, the study aims to compute the asymptotic resurgence number of a pair of graded families of ideals. To achieve this, the researchers leverage two distinct mathematical frameworks: Attouch-Wets topology and the convex-geometric properties inherent to Newton-Okounkov regions. The combination of these techniques is instrumental in establishing a characterization. Specifically, the asymptotic resurgence number is characterized through direct containment relations that exist between the pair of corresponding Newton-Okounkov regions. This indicates that the geometric arrangement and relationships of these convex regions provide a means to understand and quantify the asymptotic resurgence number.
Findings
The primary finding of this research is a characterization of the asymptotic resurgence number for a pair of graded families of ideals. This characterization is achieved by establishing a direct link to the containment relations between their respective Newton-Okounkov regions. By applying techniques derived from Attouch-Wets topology in conjunction with an analysis of the convex-geometric properties of these regions, the study demonstrates that the asymptotic resurgence number can be understood and expressed in terms of these geometric containment relationships. This establishes a connection between an algebraic invariant (asymptotic resurgence number) and geometric properties (containment of convex regions).