[Paper Review] Uniruledness of stable base loci of adjoint linear systems with and without Mori Theory

Sébastien Boucksom, Amaël Broustet+1 more|arXiv (Cornell University)|Feb 6, 2009

Algebraic Geometry and Number Theory12 references14 citations

TL;DR

This paper establishes the uniruledness of stable and augmented base loci for adjoint divisors on klt pairs using the Minimal Model Program and Takayama's extension technique. It proves that irreducible components of the non-nef, restricted, and augmented base loci of adjoint divisors $K_X + \Delta$ are uniruled when $(X, \Delta)$ is klt, extending earlier results for smooth varieties and nef canonical bundles.

ABSTRACT

We explain how to deduce from recent results in the Minimal Model Program a general uniruledness theorem for base loci of adjoint divisors. We also show how to recover special cases by extending a technique introduced by Takayama.

Motivation & Objective

• To establish uniruledness of base loci for adjoint divisors $K_X + \Delta$ on klt pairs $(X, \Delta)$ using the Minimal Model Program.
• To extend Takayama's method for proving uniruledness in the case where $-K_X$ or $L - K_X$ is nef.
• To clarify the relationship between non-nef, restricted, and augmented base loci in the context of adjoint linear systems.
• To provide a general framework for uniruledness results that applies to both big and pseudoeffective adjoint divisors.

Proposed method

• Uses the Minimal Model Program (MMP) to analyze the structure of base loci in klt pairs $(X, \Delta)$.
• Applies the definition of the stable base locus $\mathbb{B}(D)$ as the intersection of supports of $\mathbb{R}$-linearly equivalent effective divisors.
• Employs the augmented base locus $\mathbb{B}_+(D)$ and restricted base locus $\mathbb{B}_-(D)$ to study asymptotic behavior of linear series.
• Applies Takayama's extension theorem for log-pluricanonical forms to deduce uniruledness in special cases.
• Uses diophantine approximation to construct rational approximations of base locus thresholds for reduction arguments.
• Reduces the problem to verifying ampleness or nefness of auxiliary divisors via decomposition techniques in $\mathbb{Q}$-linear equivalence.

Experimental results

Research questions

• RQ1 Under what conditions are the irreducible components of the stable base locus $\mathbb{B}(K_X + \Delta)$ uniruled for a klt pair $(X, \Delta)$?
• RQ2 Can Takayama's method be generalized to prove uniruledness of base loci without assuming smoothness or vanishing of $-K_X$?
• RQ3 What is the relationship between the non-nef locus $\mathop{\rm NNef}(D)$, $\mathbb{B}_-(D)$, and $\mathbb{B}_+(D)$ for adjoint divisors?
• RQ4 Does the uniruledness of base loci persist when $(X, \Delta)$ has log-canonical, rather than klt, singularities?
• RQ5 Can the uniruledness result be extended to non-big adjoint divisors?

Key findings

• Every irreducible component of the non-nef locus $\mathop{\rm NNef}(K_X + \Delta)$ is uniruled when $(X, \Delta)$ is klt.
• For big adjoint divisors $K_X + \Delta$, every irreducible component of $\mathbb{B}_+(K_X + \Delta)$ is uniruled.
• When $X$ is smooth and either $-K_X$ or $L - K_X$ is nef, every component of $\mathbb{B}_-(L)$ is uniruled if $L$ is pseudoeffective.
• If $L$ is big and $-K_X$ or $L - K_X$ is nef, then every component of $\mathbb{B}(L)$ and $\mathbb{B}_+(L)$ is uniruled.
• The equality $\mathop{\rm NNef}(K_X + \Delta) = \mathbb{B}_-(K_X + \Delta)$ holds for klt pairs, and both loci have uniruled components.
• The results fail in general for log-canonical pairs, as shown by counterexamples in Example 6.4.