Does negative Kodaira dimension imply uniruledness? The open Nonvanishing core of the uniruledness–Kodaira-dimension conjecture

The Miyaoka–Mori numerical criterion (1986) characterizes uniruledness of complex projective varieties by the existence, through general points, of curves of negative canonical degree, and uniruledness forces the Kodaira dimension to be $-\infty$. The converse — must a smooth projective variety of negative Kodaira dimension be uniruled? — is the open core of the uniruledness–Kodaira-dimension conjecture (Kollár's Conjecture IV.1.12; Conjecture 0.1 of Boucksom–Demailly–Păun–Peternell). In characteristic $0$ it is open already in dimension $4$, where it is equivalent, via the BDPP description of the pseudo-effective cone, to the Nonvanishing statement that a pseudo-effective canonical bundle has non-negative Kodaira dimension — one of the two pillars of the abundance conjecture. In positive characteristic it is open in dimension at least $3$, and the reverse implication fails there: uniruled surfaces of general type exist. Two related questions have been settled: Sato (1993) refuted the conjectured equivalence of the numerical condition with separable uniruledness in positive characteristic, and Jovinelly–Lehmann–Riedl (2026) established the optimal Bend-and-Break degree bound. The problem posed here is to prove or disprove, in the open regimes, that negative Kodaira dimension implies uniruledness.

Background

A central theme of birational geometry is that the complexity of an algebraic variety is governed by its canonical bundle. Throughout, a variety is an integral separated scheme of finite type over an algebraically closed field $k$, and we work with smooth projective varieties; projectivity means the variety embeds in some projective space $\mathbb{P}^N$, which furnishes ample divisors — cohomology-positive integral combinations of hyperplane sections.

Uniruledness. The projective line is $\mathbb{P}^1$. A variety $X$ of dimension $n$ is uniruled if there exists a dominant rational map $\mathbb{P}^1 \times Y \dashrightarrow X$ with $\dim Y = n-1$; informally, a rational curve (the image of $\mathbb{P}^1$ under a rational map) passes through a general point of $X$. The variety is separably uniruled if such a map can be chosen so that the induced extension of function fields $k(X)/k(\mathbb{P}^1 \times Y)$ is separable. In characteristic $0$ every dominant rational map is separable, so the two notions coincide; in characteristic $p > 0$ separable uniruledness is strictly stronger, and the failure of separability is the source of most pathologies in positive characteristic.

Canonical class, Kodaira dimension, pseudo-effectivity. The canonical bundle $K_X$ is the top exterior power of the cotangent sheaf of $X$. For a curve $C \subset X$, the intersection number $(K_X \cdot C)$, the canonical degree of $C$, is the degree of the restriction of $K_X$ to $C$. The Kodaira dimension $\kappa(X) \in \{-\infty, 0, 1, \dots, n\}$ is the rate of growth of the plurigenera $h^0(X, mK_X)$ — the dimensions of the spaces of global sections of tensor powers of the canonical bundle — as $m \to \infty$; by convention $\kappa(X) = -\infty$ means $h^0(X, mK_X) = 0$ for every $m \geq 1$, that is, $X$ carries no pluricanonical forms at all. A divisor class on $X$ is pseudo-effective if it lies, inside the real vector space of divisor classes modulo numerical equivalence (two divisors are numerically equivalent when they have the same intersection number with every curve; passing to numerical equivalence forgets everything about a divisor except these numbers), in the closure of the cone spanned by effective divisors; informally, it is a limit of effective classes. A divisor is nef if it has non-negative intersection with every curve.

The Miyaoka–Mori criterion. Miyaoka and Mori (1986) gave a numerical characterization of uniruledness. Say that $X$ satisfies the numerical condition (NC) if some non-empty Zariski-open subset $U \subset X$ has the property that every closed point of $U$ lies on an irreducible curve $C$ with $(K_X \cdot C) < 0$. They proved, over the complex numbers, that a projective variety is uniruled if and only if it satisfies (NC); their method — Bend-and-Break, which degenerates a moving curve of negative canonical degree on a surface until it breaks into rational components — in fact shows that (NC) implies uniruledness over any algebraically closed field. In positive characteristic, uniruledness does not imply (NC), as Miyaoka and Mori themselves pointed out; Sato (1993) proved that separably uniruled smooth projective varieties do satisfy (NC), and constructed a smooth projective uniruled variety satisfying (NC) that is not separably uniruled, refuting the equivalence that Miyaoka and Mori had asked about.

Uniruledness and Kodaira dimension. If $X$ is separably uniruled, then $\kappa(X) = -\infty$: a pluricanonical form pulls back to a form on a separable cover $\mathbb{P}^1 \times Y$ whose restriction to a general fiber $\mathbb{P}^1$ is a section of $\omega_{\mathbb{P}^1}^{\otimes m} \cong \mathcal{O}_{\mathbb{P}^1}(-2m)$ (writing $\omega$ for the canonical bundle), hence vanishes identically, and separability forces the original form to vanish. In characteristic $0$ the converse direction is the content of a well-known conjecture: a smooth projective variety with $\kappa(X) = -\infty$ should be uniruled. This is part of the standard formulation recorded in Kollár's book (Conjecture IV.1.12) and as Conjecture 0.1 of Boucksom–Demailly–Păun–Peternell (BDPP).

The BDPP reduction to Nonvanishing. BDPP proved a duality between the pseudo-effective cone of divisors and the cone of movable curves (curves moving in families that cover the variety) on a projective manifold (a smooth complex projective variety), with the corollary that a smooth complex projective variety is uniruled if and only if $K_X$ is not pseudo-effective. Consequently, over $\mathbb{C}$ the implication '$\kappa(X) = -\infty$ implies uniruled' is equivalent to the Nonvanishing statement: if $K_X$ is pseudo-effective then $\kappa(X) \geq 0$. Nonvanishing is one of the two halves of the abundance conjecture (the other being semiampleness of the canonical class on minimal models — a line bundle is semiample when some positive tensor power of it is generated by its global sections and therefore defines a morphism to a projective space); both are pillars of the Minimal Model Program (MMP), the program that aims to decompose birational maps into elementary surgeries — flips and divisorial contractions driven by extremal rays, the $K_X$-negative edges of the closed cone of curves — and to reach, for every variety, a minimal model (a birational model on which the canonical class is nef) or a Mori-fibered model (a birational model fibered over a lower-dimensional base with ample relative anticanonical class). The conjecture $\kappa(X) = -\infty \Leftrightarrow$ uniruled is known in characteristic $0$ only in dimension at most $3$ (via the threefold Minimal Model Program, as recorded in the introduction of BDPP) and for non-algebraic Kähler threefolds — a Kähler manifold being a complex manifold carrying a Kähler form, that is, a closed positive real $(1,1)$-form — except possibly simple ones ('simple' here means not covered by positive-dimensional proper subvarieties).

Positive characteristic. In characteristic $p > 0$ the reverse implication fails outright: unirational surfaces — surfaces dominated by a rational map from projective space, a condition stronger than uniruledness — of non-negative Kodaira dimension are classical (see the discussion in the introduction of Ito–Ito–Liedtke), and Liedtke systematically constructed uniruled surfaces of general type (that is, of maximal Kodaira dimension). Ito–Ito–Liedtke proved that separably uniruled varieties have $\kappa = -\infty$, and that for surfaces the converse holds by classification, so in positive characteristic the question '$\kappa(X) = -\infty$ implies uniruled?' is open precisely in dimension at least $3$. Threefold abundance in positive characteristic is known only case by case; for instance, Zheng Xu proved the case of numerical dimension $2$ — the numerical dimension of a pseudo-effective divisor being the largest integer $k$ for which its $k$-fold positive intersection product is non-zero, an invariant bounded below by the Kodaira dimension — for log canonical threefold pairs (a threefold together with an effective rational boundary divisor, the pair having at worst the mildest singularity class used in the MMP) over a perfect field of characteristic $p > 3$.

Problem Statement

Prove or disprove that negative Kodaira dimension forces uniruledness, in the regimes where the literature leaves this open.

(A) Characteristic zero. Let $X$ be a smooth projective variety of dimension at least $4$ over an algebraically closed field of characteristic $0$, with $\kappa(X) = -\infty$. Must $X$ be uniruled? By the theorem of Boucksom–Demailly–Păun–Peternell this is equivalent to the Nonvanishing statement for smooth projective varieties: if the canonical bundle $K_X$ is pseudo-effective, then $\kappa(X) \geq 0$. This is exactly the open part (dimension at least $4$) of the uniruledness–Kodaira-dimension conjecture, recorded as Kollár's Conjecture IV.1.12 and as Conjecture 0.1 of Boucksom–Demailly–Păun–Peternell; the cases of dimension at most $3$ are known.

(B) Positive characteristic. Let $X$ be a smooth projective variety of dimension at least $3$ over an algebraically closed field of characteristic $p > 0$, with $\kappa(X) = -\infty$. Must $X$ be uniruled? Here the reverse implication is false (uniruled surfaces of general type exist), so no duality-based reduction is available; one may additionally ask, as a strictly stronger variant, whether $X$ must be separably uniruled, but the primary target is uniruledness.

The overarching objective is to determine, in each of the two regimes left open by the current literature, whether $\kappa(X) = -\infty$ implies that $X$ is uniruled — for all dimensions in the regime, over the full characteristic range, and for all smooth projective $X$ in that regime, with no additional hypotheses (such as restrictions on the fundamental group, on fibration structures, or on special classes of varieties). A proof of (A) would resolve one of the two pillars of the abundance conjecture in the Minimal Model Program; a counterexample to (A) would refute the Nonvanishing statement for smooth projective varieties.

References

1. Yoichi Miyaoka and Shigefumi Mori, 'A Numerical Criterion for Uniruledness', Annals of Mathematics (2) 124 (1986), no. 1, 65–69. DOI: 10.2307/1971387, https://doi.org/10.2307/1971387.
2. Ei-ichi Sato, 'A criterion for uniruledness in positive characteristic', Tôhoku Mathematical Journal (2) 45 (1993), 447–460. DOI: 10.2748/tmj/1178225839, https://doi.org/10.2748/tmj/1178225839.
3. Kazuhiro Ito, Tetsushi Ito, and Christian Liedtke, 'Deformations of rational curves in positive characteristic', Journal für die reine und angewandte Mathematik (Crelles Journal) 769 (2020), 55–86. DOI: 10.1515/crelle-2020-0003, https://doi.org/10.1515/crelle-2020-0003; full text of arXiv version 5 at https://arxiv.org/abs/1803.04448 (arXiv:1803.04448).
4. Eric Jovinelly, Brian Lehmann, and Eric Riedl, 'Optimal bounds in Bend-and-Break', Forum of Mathematics, Pi 14 (2026). DOI: 10.1017/fmp.2026.10033, https://doi.org/10.1017/fmp.2026.10033; full text of arXiv version 2 at https://arxiv.org/abs/2509.08065 (arXiv:2509.08065).
5. Sébastien Boucksom, Jean-Pierre Demailly, Mihai Păun, and Thomas Peternell, 'The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension', Journal of Algebraic Geometry 22 (2013), no. 2, 201–248. DOI: 10.1090/S1056-3911-2012-00574-8, https://doi.org/10.1090/S1056-3911-2012-00574-8; preprint at https://arxiv.org/abs/math/0405285 (arXiv:math/0405285).
6. Vladimir Lazić, Shin-ichi Matsumura, Thomas Peternell, Nikolaos Tsakanikas, and Zhixin Xie, 'The Nonvanishing Problem for varieties with nef anticanonical bundle', Documenta Mathematica 28 (2023), no. 6, 1393–1440. DOI: 10.4171/DM/936, https://doi.org/10.4171/DM/936; preprint at https://arxiv.org/abs/2202.13814 (arXiv:2202.13814).
7. Christian Liedtke, 'Uniruled Surfaces of General Type', Mathematische Zeitschrift 259 (2008), no. 4, 775–797. DOI: 10.1007/s00209-007-0248-7, https://doi.org/10.1007/s00209-007-0248-7; preprint at https://arxiv.org/abs/math/0608604 (arXiv:math/0608604).
8. Zheng Xu, 'Abundance for threefolds in positive characteristic when $\nu = 2$', arXiv:2307.03938 (2023; v3 of 17 April 2026 is the final version, to appear in Compositio Mathematica), https://arxiv.org/abs/2307.03938.
9. János Kollár, 'Rational Curves on Algebraic Varieties', Springer, Berlin, 1996. DOI: 10.1007/978-3-662-03276-3, https://doi.org/10.1007/978-3-662-03276-3.
10. Gilberto Bini, 'Varieties with pseudoeffective canonical divisor and their Kodaira dimension', arXiv:2402.18055 (2024, revised May 2025), https://arxiv.org/abs/2402.18055.
11. Andreas Höring, Vladimir Lazić, and Christian Lehn, 'Nonvanishing results for Kähler varieties', arXiv:2508.14634 (2025), https://arxiv.org/abs/2508.14634.