Valuations and asymptotic invariants for sequences of ideals

TL;DR

This paper investigates asymptotic invariants for sequences of ideals, demonstrating they are computed by real valuations and proposing a conjecture about the nature of valuations computing these invariants.

Contribution

It introduces the concept of asymptotic jumping numbers for graded ideal sequences and relates them to real valuations, proposing a conjecture about their quasi-monomial nature.

Findings

Every asymptotic invariant is computed by a real valuation.

The conjecture holds in dimension two.

Reduction to affine space and valuation ideals in general case.

Abstract

We study asymptotic jumping numbers for graded sequences of ideals, and show that every such invariant is computed by a suitable real valuation of the function field. We conjecture that every valuation that computes an asymptotic jumping number is necessarily quasi-monomial. This conjecture holds in dimension two. In general, we reduce it to the case of affine space and to graded sequences of valuation ideals. Along the way, we study the structure of a suitable valuation space.