Abundance for uniruled pairs which are not rationally connected

Vladimir Lazi\'c

Abundance for uniruled pairs which are not rationally connected

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper proves that a projective log canonical pair whose underlying variety is uniruled but not rationally connected has a good model, conditional on the Minimal Model Program in one dimension lower.

desk verdict A genuinely new conditional result for uniruled non-RC pairs; the one load-bearing point is a citation to the author's [LM19] that deserves careful checking. read the letter →

arxiv 1908.06945 v4 pith:TJZRMFZ6 submitted 2019-08-19 math.AG

Vladimir Lazić This is my paper · ORCID

classification math.AG MSC 14E30

keywords AbundanceconjectureMinimalModelProgramgoodmodelsuniruledvarietiesrationallyconnectedlogcanonicalpairssemiamplenessMRCfibration

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper targets the Abundance conjecture, the claim that a projective log canonical pair $(X,\Delta)$ whose canonical class $K_X+\Delta$ is pseudoeffective has a good model—a birational model in which $K_X+\Delta$ becomes semiample. It proves this for all pairs whose underlying variety $X$ is uniruled (covered by rational curves) but not rationally connected (points cannot be joined by rational curves), assuming the Minimal Model Program holds in dimension $\dim X-1$. In dimension 4, where that program is known, the conclusion is unconditional. The same method also covers pairs mapping onto non-uniruled varieties or abelian varieties, and for rationally connected $X$ it reduces the remaining problem to a sharply formulated Nonexistence Conjecture.

What carries the argument

The maximal rationally connected (MRC) fibration of $X$: a dominant rational map whose very general fibres are rationally connected and whose base is not uniruled. The proof resolves indeterminacies to make it a fibration, then splits into two cases. If a very general fibre has maximal Kodaira dimension for $K_F+\Delta|_F$, subadditivity of Kodaira dimension forces $\kappa(X,K_X+\Delta)>0$, and known abundance results for positive Kodaira dimension apply. If the fibre Kodaira dimension is smaller, a relative $(K_X+\Delta)$-MMP over the base terminates in a relative good model, and the canonical bundle formula transfers abundance to the base. For real coefficients, the argument passes through a decomposition of the divisor into finitely many rational pieces, and the non-klt part is handled by perturbing the boundary.

What would settle it

Construct a projective log canonical fourfold $X$ that is uniruled but not rationally connected with $K_X+\Delta$ nef but not semiample; since the Minimal Model Program is known in dimension 4, such a pair would refute the unconditional Corollary B.

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Extended reading notes

Core claim

Theorem A is the central claim: assuming good models exist for non-uniruled klt pairs with rational boundaries in dimension $n-1$, every projective log canonical pair $(X,\Delta)$ of dimension $n$ with $X$ uniruled but not rationally connected and $K_X+\Delta$ pseudoeffective has a good model. In particular, if $K_X+\Delta$ is nef, then it is semiample. The stronger Theorem C replaces the MRC fibration by any dominant rational map to a normal projective variety $Y$ with $0<\dim Y<\dim X$ and $Y$ not uniruled, and Corollary 3.2 applies when the target is an abelian variety. For rationally connected $X$, Theorem D shows the same conclusion follows from an explicit Nonexistence Conjecture for special klt pairs of Calabi-Yau type, and Theorem 4.1 makes it unconditional when $\kappa(X,\Delta)>0$ or $\kappa(X,-K_X)>0$.

Load-bearing premise

The proof rests on the assumption that every non-uniruled klt pair with rational boundary in dimension $n-1$ has a good model, and for the rationally connected case also on an unproved Nonexistence Conjecture for special klt pairs of Calabi-Yau type.

Editorial extensions

If this is right

In dimension 4, every projective log canonical pair with $K_X+\Delta$ pseudoeffective and $X$ uniruled but not rationally connected has a good model; if $K_X+\Delta$ is nef, it is semiample.
A projective log canonical pair of dimension $n$ with a dominant rational map onto a non-uniruled variety of intermediate dimension has a good model, under the same lower-dimensional MMP assumption.
A projective log canonical pair with a nontrivial rational map to an abelian variety has a good model, under the same assumption.
For rationally connected $X$, the existence of a good model follows from the paper's Nonexistence Conjecture; if either $\kappa(X,\Delta)>0$ or $\kappa(X,-K_X)>0$, the good model exists without that conjecture.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The reduction to the base of the MRC fibration suggests that abundance for uniruled varieties may be governed by the Kodaira dimension of a non-uniruled base, so one testable extension is to replace rational connectedness by other fibration properties that force the base to have pseudoeffective canonical class.
The proof of Theorem D converts abundance for rationally connected pairs into a statement about nef divisors on klt pairs of Calabi-Yau type; proving the Nonexistence Conjecture would remove the last conditional assumption in the rationally connected case.
Because all ingredients are known in dimension 3, the paper's dimension-4 result is unconditional; a concrete check would be to run the same argument on explicit 4-fold examples to see whether nef non-semiample canonical divisors are ruled out.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear X,

Here's my take on Lazi'c's Abundance paper. The headline: it's a serious, short paper that proves Abundance for uniruled but not rationally connected lc pairs, conditional on the MMP in one dimension lower. The main new ingredient is the MRC fibration step: either the fibre has non-maximal Kodaira dimension and you run a relative MMP, or it has maximal Kodaira dimension and subadditivity contradicts kappa = 0. That's genuinely new; [DL15] only got (X, tau Delta).

The paper is also honest about what it does not do. Theorem D for the rationally connected case is conditional on a nonexistence conjecture, and the author says so. The real-coefficient case is handled by decomposing the boundary, a standard but delicate step, and it's done correctly. Corollary B gives an unconditional dimension-4 result, which is a nice concrete payoff.

Now the soft spots. The stress-test note has a point: Step 2 of Theorem A needs a good model for the general fibre F of the MRC fibration, which is rationally connected. The stated hypothesis only gives good models for non-uniruled klt pairs with rational boundaries in dimension n-1. The proof bridges this with one sentence: 'By [LM19, Theorem 1.3 and Lemmas 2.3 and 2.4] we may assume the existence of good models for log canonical pairs in dimensions at most n-1.' That upgrade is doing real work. I think it's plausible - uniruled klt pairs are easier, and [LM19] is precisely about nonvanishing and good models there - but the paper doesn't reproduce it, and [LM19] is the author's own. A referee should check that citation line carefully. If the upgrade fails, Step 2 collapses. Everything else in the proof is built from cited results with clean logic.

There's also the general point that much of the proof sits on recent deep results (KP17, LT19, LM19). That's normal in this field, but it means the paper's correctness is only as solid as that network. I don't see circularity or overclaim.

My bottom line: this deserves serious peer review. The central theorem is a real advance, the proof is a nice idea, and the one fragile spot is checkable by a knowledgeable referee. I'd send it on, and ask the referee to verify the [LM19] reduction. I would cite it myself.

Referee Report

0 major / 4 minor

Summary. The paper proves that a projective log canonical pair (X,Δ) of dimension n whose underlying variety X is uniruled but not rationally connected has a good model whenever K_X+Δ is pseudoeffective, assuming the existence of good models for non-uniruled klt pairs with rational boundaries in dimension n−1. The proof uses the MRC fibration, the nonvanishing theorem for uniruled log canonical pairs, subadditivity of Kodaira dimension, and a reduction to lower-dimensional cases via recent work of the author and others. The paper also proves a generalisation (Theorem C) for pairs admitting a dominant rational map to a non-uniruled base, and a conditional result (Theorem D) for rationally connected pairs subject to a newly formulated Nonexistence Conjecture.

Significance. The main theorems provide a substantial advance on the abundance conjecture for a large class of uniruled pairs, going beyond previous results that required fibrations over abelian varieties or other special structures. The paper is carefully written and explicitly separates the new arguments from the quoted machinery. The reduction of the rationally connected case to a very specific Nonexistence Conjecture is a useful conceptual contribution, and the paper also gives a clean proof for rationally connected pairs with κ(X,Δ)>0 or κ(X,−K_X)>0. The proofs are coherent and the conditional hypotheses are stated precisely.

minor comments (4)

[Section 3, proof of Theorem A, opening sentence] The assertion that by [LM19, Theorem 1.3 and Lemmas 2.3 and 2.4] one may assume the existence of good models for all log canonical pairs in dimensions at most n−1 is essential for Step 2, where it is applied to the rationally connected general fibre F of the MRC fibration. Since the stated hypothesis of Theorem A covers only non-uniruled klt pairs with rational boundaries, it would be helpful to state the precise result from [LM19] that justifies this strengthening, or to give a short indication of the derivation.
[Section 3, Step 2 of Theorem A] The sentence 'Since K_F+Δ|F is pseudoeffective, we have κ(F, K_F+Δ|F) ≥ 0 by the assumption in lower dimensions' is slightly imprecise, because the lower-dimensional assumption is about existence of good models rather than about nonvanishing. Clarifying that this follows from the upgraded assumption introduced at the start of the proof would avoid potential confusion.
[Section 4, Step 4 of Theorem D] The parameter δ in the displayed relation K_X+δΔ ∼Q 0 is not explicitly defined. For clarity, one should note that δ = 1 − d_1/δ_1, which is indeed between 0 and 1 in the situation at hand.
[Throughout] Some equations and references are cited without precise theorem numbers in a few places (e.g., the use of [KP17, Theorem 9.9] and [Nak04, Corollary V.1.12]); adding the exact statements would improve the reader's ability to verify the arguments.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are conditional reductions to lower-dimensional MMP/good models, with explicit external and separate hypotheses.

full rationale

The derivation chain of Theorems A and C is a standard induction: under the explicit assumption of good models for non-uniruled klt pairs in dimension n-1, the MRC fibration and known results ([KP17], [HX13], [LT19], [Lai11], [Has19]) are used to compare the invariant Kodaira dimension with the numerical dimension. The paper repeatedly invokes [LM19] (same author) at the start of proofs: 'By [LM19, Theorem 1.3 and Lemmas 2.3 and 2.4] we may assume the existence of good models for log canonical pairs in dimensions at most n-1.' This self-citation is load-bearing, for example Step 2 of Theorem A needs a good model for the rationally connected fibre F, but it is not circular within the present paper: [LM19] is a prior external theorem with its own assumptions, which do not include Theorems A or C. Whether that cited upgrade is fully proved is a correctness or verifiability question, not a circularity one. Similarly, the Nonexistence Conjecture used in Theorem D is stated as an explicit separate hypothesis, and the paper even notes that the conjecture is implied by the Abundance conjecture; it is not a disguised version of the theorem being proved. No fitted parameter is renamed as a prediction, and no displayed equation is defined in terms of the claimed conclusion. Hence the paper exhibits no circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper is a theorem-proving work in birational geometry. It introduces no new mathematical objects, only a conditional statement and a new conjecture. The central claim rests on the MMP in lower dimensions and on several recent published theorems. The Nonexistence Conjecture is an ad-hoc conjecture for the rationally connected case, clearly flagged as such.

assumptions (6)

domain assumption Existence of good models for non-uniruled klt pairs with rational boundaries in dimension n-1.
The explicit hypothesis of Theorems A, C, and D, used throughout to obtain lower-dimensional good models. It is a standard open conjecture in the MMP, known up to dimension 3.
standard math BDPP theorem: a smooth projective variety is not uniruled if and only if its canonical divisor is pseudoeffective.
Used in Step 1 of Theorem A to show K_Y is pseudoeffective for the base of the MRC fibration, and in the rationally connected case to show K_X is not pseudoeffective. Cited as [BDPP13, Corollary 0.3].
standard math Subadditivity of Kodaira dimension for fibrations (Kovacs-Patakfalvi, Theorem 9.9).
Used in Step 2 of Theorem A to derive a contradiction when the fibre Kodaira dimension is maximal, by comparing Kodaira dimensions of X, Y, and the fibre. Cited as [KP17, Theorem 9.9].
standard math Nonvanishing for uniruled log canonical pairs (Lazic-Meng, Theorem 1.1).
Used via Theorem 2.2(a) to show kappa(X,K_X+Delta) >= 0 for uniruled pairs. Cited as [LM19, Theorem 1.1].
standard math Existence of minimal models under various conditions (Lazic-Tsakanikas, Theorems B, C, E, F).
Used in Steps 3 of Theorems A and C to obtain minimal models for pairs with real boundaries, and in Theorem D Step 4 for termination of the MMP. Cited as [LT19].
ad hoc to paper Nonexistence Conjecture: there does not exist a klt pair (X,Delta) with X rationally connected, Delta a nef Q-divisor with prime support, kappa(X,Delta)=0, K_X+Delta ~Q 0, and Delta.C > 0 for every curve through a very general point.
Introduced in Section 4 and required for Theorem D. The author notes it follows from the Abundance conjecture, so it is not an established result. It is used in Step 5 of the proof of Theorem D to rule out the case when the nef reduction is trivial.

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Pith. "Pith review of Abundance for uniruled pairs which are not rationally connected." pith.science (2026). https://pith.science/paper/TJZRMFZ6

@misc{pith2026190806945,
  author       = {Pith},
  title        = {Pith review of: Abundance for uniruled pairs which are not rationally connected},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TJZRMFZ6}},
  note         = {Machine review of arXiv:1908.06945}
}

abstract

One of the central aims of the Minimal Model Program is to show that a projective log canonical pair $(X,\Delta)$ with $K_X+\Delta$ pseudoeffective has a good model, i.e.\ a minimal model $(Y,\Delta_Y)$ such that $K_Y+\Delta_Y$ is semiample. The goal of this paper is to show that this holds if $X$ is uniruled but not rationally connected, assuming the Minimal Model Program in dimension $\dim X-1$. Moreover, if $X$ is rationally connected, then we show that the existence of a good minimal model for $(X,\Delta)$ follows from a nonexistence conjecture for a very specific class of rationally connected pairs of Calabi--Yau type.

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Ambro, The moduli b -divisor of an lc-trivial fibration , Compos

F. Ambro, The moduli b -divisor of an lc-trivial fibration , Compos. Math. 141 (2005), no. 2, 385--403

work page 2005

[2] [2]

, A semiampleness criterion, J. Math. Sci. Univ. Tokyo 12 (2005), no. 3, 445--466

work page 2005

[3] [3]

Birkar and J

C. Birkar and J. A. Chen, Varieties fibred over abelian varieties with fibres of log general type, Adv. Math. 270 (2015), 206--222

work page 2015

[4] [4]

Bauer, F

Th. Bauer, F. Campana, Th. Eckl, S. Kebekus, Th. Peternell, S. Rams, T. Szemberg, and L. Wotzlaw, A reduction map for nef line bundles, Complex geometry ( G \"ottingen, 2000), Springer, Berlin, 2002, pp. 27--36

work page 2000

[5] [5]

Boucksom, J.-P

S. Boucksom, J.-P. Demailly, M. P a un, and Th. Peternell, The pseudo-effective cone of a compact K \"ahler manifold and varieties of negative K odaira dimension , J. Algebraic Geom. 22 (2013), no. 2, 201--248

work page 2013

[6] [6]

Birkar, On existence of log minimal models II , J

C. Birkar, On existence of log minimal models II , J. reine angew. Math. 658 (2011), 99--113

work page 2011

[7] [7]

S. R. Choi, The geography of log models and its applications , PhD Thesis, Johns Hopkins University, 2008 0= 2008

work page 2008

[8] [8]

Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001

O. Debarre, Higher-dimensional algebraic geometry, Universitext, Springer-Verlag, New York, 2001

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