Pseudo-effective Divisors Research Articles

Local Numerical Equivalences and Okounkov Bodies in Higher Dimensions

We continue to explore the numerical nature of the Okounkov bodies focusing on the local behaviors near given points. More precisely, we show that the set of Okounkov bodies of a pseudoeffective divisor with respect to admissible flags centered at a fixed point determines the local numerical equivalence class of divisors, which is defined in terms of refined divisorial Zariski decompositions. Our results extend Roé’s work [R] on surfaces to higher-dimensional varieties although our proof is essentially different in nature.

FINITENESS OF LOG MINIMAL MODELS AND NEF CURVES ON -FOLDS IN CHARACTERISTIC

In this article, we prove a finiteness result on the number of log minimal models for 3-folds in $\operatorname{char}p>5$. We then use this result to prove a version of Batyrev’s conjecture on the structure of nef cone of curves on 3-folds in characteristic $p>5$. We also give a proof of the same conjecture in full generality in characteristic 0. We further verify that the duality of movable curves and pseudo-effective divisors hold in arbitrary characteristic. We then give a criterion for the pseudo-effectiveness of the canonical divisor $K_{X}$ of a smooth projective variety in arbitrary characteristic in terms of the existence of a family of rational curves on $X$.

Okounkov bodies associated to pseudoeffective divisors

Okounkov bodies associated to pseudoeffective divisors

Numerical Dimension and Locally Ample Curves

In the paper [the author, “Fujita vanishing theorems for q -ample divisors and applications on subvarieties with nef normal bundle”, Preprint, arXiv:1609.07797], was shown that the restriction of a pseudoeffective divisor D to a so-called nef subvariety Y (e.g. Y is lci in X and has nef normal bundle) is pseudoeffective. Assuming the normal bundle is ample and that D|_Y is not big, we prove that the numerical dimension of D is bounded above by that of its restriction, i.e. \kappa_{\sigma}(D)\leq \kappa_{\sigma}(D|_Y) . The main motivation is to study the cycle classes of “positive” curves: we show that the cycle class of a curve with ample normal bundle lies in the interior of the cone of curves, and the cycle class of an ample curve lies in the interior of the cone of movable curves. We do not impose any condition on the singularities on the curve or the ambient variety. For locally complete intersection curves in a smooth projective variety, this is the main result of J. C. Ottem [J. Eur. Math. Soc. (JEMS) 18, No. 11, 2459–2468 (2016; Zbl 1362.14007)]. The main tool in this paper is the theory of q -ample divisors.

Okounkov Bodies Associated to Pseudoeffective Divisors II

We first prove some basic properties of Okounkov bodies and give a characterization of Nakayama and positive volume subvarieties of a pseudoeffective divisor in terms of Okounkov bodies. Next, we show that each valuative and limiting Okounkov bodies of a pseudoeffective divisor which admits the birational good Zariski decomposition is a rational polytope with respect to some admissible flag. This is an extension of the result of Anderson-Kuronya-Lozovanu about the rational polyhedrality of Okounkov bodies of big divisors with finitely generated section rings.

Numerical analogues of the Kodaira dimension and the Abundance Conjecture

We add further notions to Lehmann’s list of numerical analogues to the Kodaira dimension of pseudo-effective divisors on smooth complex projective varieties, and show new relations between them. Then we use these notions and relations to fill in a gap in Lehmann’s arguments, thus proving that most of these notions are equal. Finally, we show that the Abundance Conjecture, as formulated in the context of the Minimal Model Program, and the Generalized Abundance Conjecture using these numerical analogues to the Kodaira dimension, are equivalent for non-uniruled complex projective varieties.

A pathology of asymptotic multiplicity in the relative setting

We point out an example of a projective family $\pi : X \to S$, a $\pi$-pseudoeffective divisor $D$ on $X$, and a subvariety $V \subset X$ for which the asymptotic multiplicity $\sigma_V(D;X/S)$ is infinite. This shows that the divisorial Zariski decomposition is not always defined for pseudoeffective divisors in the relative setting.

Rationality of Euler–Chow series and finite generation of Cox rings

We study the rationality of the Euler–Chow series E1(X) of codimension one cycles on a projective variety X and its relation with the effective cone and the Cox ring of X. Among other things, we prove that E1(X) is transcendental if the cone NE1(X) of pseudo-effective divisors on X has infinitely many extremal rays generated by effective divisors. On the other hand, we give examples showing that the converse fails. In addition, we give an example where E1(X) is rational and Cox(X) is infinitely generated. Finally, we compute E1(X) for Del Pezzo surfaces X.

MINKOWSKI DECOMPOSITION AND GENERATORS OF THE MOVING CONE FOR TORIC VARIETIES:

We prove that for smooth projective toric varieties, the Okounkov body of a |$T$|-invariant pseudo-effective divisor with respect to a |$T$|-invariant flag decomposes as a finite Minkowski sum of indecomposable polytopes, and that the set of these polytopes corresponds to a finite Minkowski basis whose elements span the extremal rays in the secondary fan. In fact, the Minkowski basis does not depend on the choice of the |$T$|-invariant flag. Moreover, we present an algorithm that computes the Minkowski basis.

Numerical triviality and pullbacks

Let f:X→Z be a surjective morphism of smooth complex projective varieties with connected fibers. Suppose that L is a pseudo-effective divisor that is f-numerically trivial. We show that there is a divisor D on Z such that L≡f⁎D.

On the existence of a weak Zariski decomposition on projectivized vector bundles

A pseudoeffective divisor is said to have a weak Zariski decomposition if it can be, up to a birational transformation, numerically written as the sum of a nef and an effective divisor. In this paper we consider the problem of the existence of a weak Zariski decomposition for each pseudoeffective divisor on a variety \(X= \mathbb {P}(\fancyscript{E})\), where \(\fancyscript{E}\) is a vector bundle on a smooth complex projective variety \(Z\) of Picard number one. We prove the existence of such a decomposition in a number of meaningful situations.

On the Numerical Dimension of Pseudo-Effective Divisors in Positive Characteristic

Let $X$ be a smooth projective variety over an algebraically closed field of positive characteristic. We prove that if $D$ is a pseudo-effective ${\bf R}$-divisor on $X$ which is not numerically equivalent to the negative part in its divisorial Zariski decomposition, then the numerical dimension of $D$ is positive. In characteristic zero, this was proved by Nakayama using vanishing theorems.

Algebraic bounds on analytic multiplier ideals

Given a pseudo-effective divisor L we construct the diminished ideal of L, a continuous extension of the asymptotic multiplier ideal for big divisors to the pseudo-effective boundary. For most pseudo-effective divisors L the multiplier ideal of the metric of minimal singularities of L is contained in the diminished ideal. We also characterize abundant divisors using the diminished ideal, indicating that in this case the geometric and analytic information should coincide.

Comparing numerical dimensions

The numerical dimension is a numerical measure of the positivity of a pseudoeffective divisor L. There are several proposed definitions of the numerical dimension due to Nakayama and Boucksom et al. We prove the equality of these notions and give several additional characterizations. We also prove some new properties of the numerical dimension.

On Eckl’s pseudo-effective reduction map

Suppose that X is a complex projective variety and L is a pseudo-effective divisor. A numerical reduction map is a quotient of X by all subvarieties along which L is numerically trivial. We construct two variants: the L-trivial reduction map and the pseudo-effective reduction map of [Eckl05]. We show that these maps capture interesting geometric properties of L and use them to analyze abundant divisors.

Asymptotic base loci on singular varieties

We prove that the non-nef locus and the restricted base locus of a pseudoeffective divisor coincide on KLT pairs. We also extend to KLT pairs F. Russo’s characterization of nef and abundant divisors by means of asymptotic multiplier ideals.

The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a duality statement between the cone of pseudo-effective divisors and the cone of “movable curves”, which is obtained from a general theory of movable intersections and approximate Zariski decomposition for closed positive ( 1 , 1 ) (1,1) -currents. As a corollary, a projective manifold has a pseudo-effective canonical bundle if and only if it is not uniruled. We also prove that a 4-fold with a canonical bundle which is pseudo-effective and of numerical class zero in restriction to curves of a good covering family, has non-negative Kodaira dimension.

The cone of pseudo-effective divisors of log varieties after Batyrev

In these notes we investigate the cone of nef curves of projective varieties, which is the dual cone to the cone of pseudo-effective divisors. We prove a structure theorem for the cone of nef curves of projective $\mathbb Q$-factorial klt pairs of arbitrary dimension from the point of view of the Minimal Model Program. This is a generalization of Batyrev's structure theorem for the cone of nef curves of projective terminal threefolds.

Adjoint Linear Systems on Normal Log Surfaces

We prove a Reider type theorem for separating any cluster by an adjoint system to a pseudoeffective divisor on a normal surface. As a corollary we get a Reider type theorem for adjoint linear systems (to a nef -divisor) on normal log surfaces. This theorem is new even for smooth surfaces.