[Paper Review] Asymptotic invariants of base loci

Lawrence Ein, Robert Lazarsfeld+3 more|ArXiv.org|Aug 12, 2003

Algebraic Geometry and Number Theory12 references4 citations

TL;DR

This paper introduces and studies asymptotic invariants of base loci for big divisors on projective varieties, focusing on the asymptotic order of vanishing along discrete valuations. It establishes that these invariants depend only on numerical equivalence classes and vary continuously, extending previous results on base loci and multiplier ideals, with applications to volume functions and piecewise polynomial behavior on Mori dream spaces.

ABSTRACT

The purpose of this paper is to define and study systematically some asymptotic invariants associated to base loci of line bundles on smooth projective varieties. We distinguish an open dense subset of the real big cone, called the stable locus, consisting of the set of classes on which the asymptotic base locus is locally constant. The asymptotic invariants define continuous functions on the big cone, whose vanishing characterizes, roughly speaking, the unstable locus. We show that for toric varieties at least, there exists a polyhedral decomposition of the big cone on which these functions are polynomial.

Motivation & Objective

• To define and systematically study asymptotic invariants associated with base loci of linear series on projective varieties.
• To understand how these invariants—particularly the asymptotic order of vanishing—behave under numerical equivalence and perturbations by ample divisors.
• To extend results of Nakayama and others on $σ_E(D)$ to general discrete valuations and real divisor classes.
• To analyze the structure of augmented and restricted base loci, showing they depend only on numerical classes.
• To establish piecewise polynomial behavior of volume functions under finite generation assumptions on linear series.

Proposed method

• Define the asymptotic order of vanishing $v(\|D\|) = \lim_{p \to \infty} \frac{v(|pD|)}{p}$ for big $\mathbb{Q}$-divisors $D$ and discrete valuations $v$.
• Prove that $v(\|D\|)$ depends only on the numerical equivalence class of $D$, extending to a continuous function on $\operatorname{Big}(X)_{\mathbb{R}}$.
• Introduce the augmented base locus $\mathbf{B}_+(D) = \bigcap_A \mathbf{B}(D - A)$ and restricted base locus $\mathbf{B}_{--}(D) = \bigcup_A \mathbf{B}(D + A)$ over ample $A$.
• Show that $\mathbf{B}_+(D)$ and $\mathbf{B}_{--}(D)$ depend only on the numerical class of $D$, enabling extension to real classes.
• Use Fujita's Approximation Theorem and birational models to express volume via $(L^{[n]}) = (M^n)$ for $M = \pi^*L - F$.
• Apply fan refinements of the closed big cone $\overline{\operatorname{Big}(X)}_{\mathbb{R}}$ to prove piecewise polynomiality of volume and asymptotic invariants.

Experimental results

Research questions

• RQ1 How do asymptotic invariants like $v(\|D\|)$ behave as functions of the numerical class of a big divisor $D$?
• RQ2 Can the stable base locus $\mathbf{B}(D)$ be replaced by numerically well-behaved approximations like $\mathbf{B}_+(D)$ and $\mathbf{B}_{--}(D)$?
• RQ3 Under what conditions is the volume function $\operatorname{vol}(L)$ piecewise polynomial on the big cone?
• RQ4 To what extent can asymptotic invariants be defined and computed via linear series on birational models?
• RQ5 What is the relationship between the center of a valuation $v$ and the non-vanishing of $v(\|\xi\|)$ for $\xi \in \operatorname{Big}(X)_{\mathbb{R}}$?

Key findings

• The asymptotic order of vanishing $v(\|D\|)$ depends only on the numerical equivalence class of $D$, and extends uniquely to a continuous function on $\operatorname{Big}(X)_{\mathbb{R}}$.
• $\mathbf{B}_+(D)$ and $\mathbf{B}_{--}(D)$ are well-defined for real classes $\xi \in \operatorname{Big}(X)_{\mathbb{R}}$ and depend only on numerical equivalence.
• For a smooth variety $X$, $v(\|\xi\|) > 0$ if and only if the center $Z$ of $v$ is contained in $\mathbf{B}_{--}(\xi)$.
• When $X$ has finitely generated linear series, the volume function $\operatorname{vol}(L)$ is piecewise polynomial with respect to a fan refinement of $\overline{\operatorname{Big}(X)}_{\mathbb{R}}$.
• The asymptotic invariant $\widetilde{\operatorname{e}_Z}(m)$ is a polynomial of degree $d$ on each cone in a fan refinement of the big cone, under suitable conditions.
• The volume satisfies $\operatorname{vol}(L) = \sup_m \frac{((mL)^{[n]})}{m^n}$, where $(L^{[n]})$ is defined via a birational model resolving the base ideal of $L$.