Divisorial Mori contractions of submaximal length

Bruno Dewer

REVIEW 3 major objections 4 minor 18 references

Divisorial Mori contractions of submaximal length

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For a birational divisorial elementary Mori contraction of submaximal length, the exceptional locus is birational either to a projective bundle or to a quadric bundle.

desk verdict A meaningful submaximal analogue of Höring–Novelli, but the central dichotomy rests on a deformation estimate that fails on the singular fibres the paper explicitly allows. read the letter →

arxiv 2411.17549 v1 pith:GBI36LML submitted 2024-11-26 math.AG

Bruno Dewer This is my paper · ORCID

classification math.AG MSC 14E3014J4514D06

keywords divisorialMoricontractionsubmaximallengthexceptionallocusprojectivebundlequadricdeformationfamilyofrationalcurvesFanomanifoldbirationalgeometry

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

This paper studies birational divisorial elementary Mori contractions whose length is one less than maximal, i.e., $l(f)=n-1$ where $n$ is the dimension of the general fibre of the exceptional divisor. It proves that the equidimensional part of the exceptional locus is birational either to a family of projective spaces or to a quadric bundle. In the projective-space case, for even $n$ the family is actually the projectivization of a vector bundle. In the quadric case, reducible fibres have exactly two components, each normalized by projective space. The result extends the known maximal-length statement (projective bundle) and gives a relative analogue of the absolute characterizations of projective spaces and quadrics.

What carries the argument

The central objects are the length $l(f)=n-1$ (the minimal degree of $-K_X$ on contracted rational curves) and the $f$-minimal curves realizing this degree. Lemmas 3.1 and 3.2 show that these curves form an unsplit, surjective deformation family covering the equidimensional locus. The dichotomy $-E\cdot\Gamma\in\{1,2\}$ comes from comparing the bend-and-break lower bound $\dim H\ge 2n-4-E\cdot\Gamma$ with the upper bound $\dim H\le 2n-2$ supplied by the Cho--Miyaoka--Shepherd-Barron characterization of projective space. The conductor divisor (Lemma 2.5) then forces the fibre structure, and the surjectivity of the relative evaluation map (Theorem 2.7, due to Andreatta and Wiśniewski) builds the projective or quadric bundle model.

What would settle it

A concrete observation that would settle the claim: find a birational divisorial elementary Mori contraction satisfying Setup 1.4 with an $n$-dimensional fibre over a smooth point of $Z$ containing no $f$-minimal curve, or an $f$-minimal curve $\Gamma$ with $-E\cdot\Gamma\ge 3$; either outcome would refute the dichotomy, and a computation showing Lemma 3.1's surjectivity fails would also falsify the main theorem.

Watch

Extended reading notes

Core claim

Under Setup 1.4, for any $f$-minimal curve $\Gamma$ the intersection number $-E\cdot\Gamma$ can only be $1$ or $2$. When it is $2$, every $n$-dimensional fibre of $E_{\mathrm{eq}}\to Z_{\mathrm{eq}}$ is normalized by $\mathbb{P}^n$ and the whole fibration is birational to a family of projective spaces; for $n$ even it is isomorphic to the projectivization of a vector bundle over $Z_{\mathrm{eq}}$. When it is $1$, the fibration is birational to a quadric bundle, every reducible fibre has two irreducible components whose reductions are normalized by $\mathbb{P}^n$, irreducible generically reduced fibres are quadrics, and nonreduced fibres have reduction $\mathbb{P}^n$.

Load-bearing premise

The whole dichotomy rests on the claim in Lemmas 3.1 and 3.2 that the $f$-minimal curves form an unsplit deformation family that covers every $n$-dimensional fibre of the exceptional divisor; if some fibre lacked a minimal curve, or the family degenerated into reducible curves, the dichotomy $-E\cdot\Gamma=1,2$ would collapse.

Editorial extensions

If this is right

The equidimensional part of the exceptional divisor of a submaximal divisorial Mori contraction is always birational to a $\mathbb{P}^n$-bundle or a quadric bundle.
When $-E\cdot\Gamma=2$ and $n$ is even, the birational modification can be removed: $E_{\mathrm{eq}}$ is the projectivization of a vector bundle over $Z_{\mathrm{eq}}$.
In the quadric-bundle case, reducible fibres have exactly two components (each normalized by $\mathbb{P}^n$) and nonreduced fibres have reduction $\mathbb{P}^n$, so the local structure of the exceptional divisor is completely pinned down.
Together with the maximal-length theorem, this gives a relative dichotomy: maximal length forces projective bundles; submaximal length forces projective or quadric bundles.

Reading between the lines

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

The generic fibre dichotomy (projective space or quadric) mirrors the absolute pseudoindex characterization of Fano manifolds, suggesting that the length deficit $n-1$ is the relative avatar of pseudoindex $n$.
The parity obstruction for $n$ suggests that whether the family is globally projectivized may depend on the vanishing of an even-degree cohomology class (e.g., a Brauer-type invariant); for odd $n$ the paper leaves open whether non-projectivized families with $-E\cdot\Gamma=2$ exist.
The §5 example indicates that dropping equidimensionality creates special fibres isomorphic to $\mathbb{P}^3$ inside a quadric-bundle family, so a full global statement would need to track non-equidimensional fibres separately, presumably as additional blow-ups.
The same degeneration-of-minimal-curves technique could be applied to contractions of length $n-k$ for $k>1$, plausibly yielding a hierarchy of fibre models (complete intersections of quadrics, etc.), though the paper stops at $k=1$.

Share X Bluesky LinkedIn Reddit HN

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

Read Dewer's paper. The intended result is worth having: a submaximal-length divisorial Mori contraction whose exceptional locus is birational to a projective bundle or a quadric bundle, with a clean projective-space versus quadric dichotomy and an explicit non-equidimensional example. The organization is good, the Section 4 constructions are plausible, and the citation pattern is appropriate—the paper really is extending [HN13] rather than repackaging it. No circularity, no fitted parameters.

The soft spot is load-bearing. The proofs of Lemma 3.1 and Proposition 3.3 use [Kol96, II.1.3] to lower-bound deformation spaces of rational curves in the fibres of E→Z. That bound is a smooth-target deformation estimate. The fibres are only lci and may be singular; the classification later explicitly allows normal quadrics, including quadric cones. On an n-dimensional quadric cone (n≥3), the ruling lines meet the smooth locus, but their deformation family has dimension n−1, not the claimed −K_Q·ℓ+n−3 = 2n−3. So the estimate is false in exactly the kind of fibre the theorem permits. Lemma 3.1 uses this bound to rule out −K_Ez·C > n+1 and to produce f-minimal curves in every fibre; Proposition 3.3 uses it again to force −E·Γ∈{1,2}. Without a freeness or smoothness condition on the curve, the dichotomy is not secured as written.

The rest of the paper is in better shape. The projectivization and quadric-bundle constructions in Section 4 are coherent given the dichotomy. The §5 example is instructive but the semicontinuity step proving that ε is elementary is too terse and should be expanded, though that is secondary.

Who gets value from this? Specialists in Mori theory and Fano classification. The result is likely true and the gap may be repairable—for instance by adding a suitable hypothesis on the minimal curves or by a more careful deformation-theoretic argument for non-free curves. I would not cite it as established until that is fixed, but I would want a referee to see it. Recommend sending to peer review, with a referee who knows singular deformation theory.

Referee Report

3 major / 4 minor

Summary. The paper proposes a relative numerical characterization of divisorial elementary Mori contractions of submaximal length. Under Setup 1.4, where f:X→Y is birational, divisorial, elementary, and has length l(f)=n−1, the main theorem (Theorem 1.5) claims that for an f-minimal curve Γ in the equidimensional locus Eeq one has either −E·Γ=1 or −E·Γ=2. In the first case it asserts that Eeq→Zeq is birational to a quadric bundle with reducible fibres having two components normalized by P^n; in the second case it asserts that all fibres are normalized by P^n and, for n even, that Eeq→Zeq is a projective bundle. The proof is built on Lemmas 3.1 and 3.2, which supply f-minimal curves in every fibre and show that they cover Eeq, and on a case analysis in Section 4. Section 5 constructs a nonequidimensional example of a divisorial elementary Mori contraction of submaximal length whose exceptional divisor is generically a quadric bundle with one projective-space fibre.

Significance. If the main theorem is correct, it is a natural and valuable extension of the Höring–Novelli treatment of maximal length to the submaximal case, and it fits into the line of Cho–Miyaoka–Shepherd-Barron, Kebekus, and Dedieu–Höring. The paper is clearly structured and the birational constructions in Section 4 are coherent. The author also provides an explicit example in Section 5, which is useful for understanding why the equidimensional locus is needed. However, the central proof relies on deformation-theoretic estimates for singular fibres, and these estimates are not justified in the text; since the dichotomy −E·Γ∈{1,2} and the subsequent fibre descriptions depend on them, the main theorem is not yet established as written.

major comments (3)

[Lemma 3.1] The proof applies [Kol96, Prop. II.1.3] to a fibre Ez that is only assumed to be a generically reduced local complete intersection in the smooth variety X. The cited deformation estimate is a smooth-target statement; indeed, Lemma 3.2 explicitly says 'As X is smooth, we may apply [Kol96, Theorem II.1.3]', while Lemma 3.1 suppresses the smoothness issue. On a singular fibre the estimate cannot be used as a black box: for an n-dimensional quadric cone Q with n≥4, the family of lines has dimension n−1, whereas the formula −KEz·C+n−3 would give at least 2n−4. The line does not satisfy the numerical hypothesis −KEz·C>n+1 of the subsequent contradiction, so this example does not by itself disprove Lemma 3.1, but it shows that the cited estimate is false for arbitrary singular lci fibres and that the proof needs an additional argument controlling the minimal curve C inside a singular fibre. This point is load-bearing because Lemma 3.1 supplies the f-minimal curves used in Lemma 3.2, Proposition 3.3, and the whole dichotomy −E·Γ∈{1,2}.
[Proposition 3.3] The proof invokes [CMSB02, Theorem 0.1] to bound the dimension of an unsplit deformation family in a possibly singular fibre F by 2n−2 and to infer that equality of the bound implies that F is normalized by P^n. The cited theorem is a statement about smooth projective varieties, and the text does not explain how to reduce to the normalization of F, which need not be smooth. The same use of [CMSB02, Theorem 0.1] appears again in Lemma 4.2 for a component D1 of a reducible or nonreduced fibre. Without a singular analogue of this dimension bound, or a separate proof, the claimed dichotomy and the fibre descriptions in Theorem 1.5 are not established.
[Proposition 3.3, case (ii)] The sentence 'we may conclude that Ez is isomorphic to a quadric' is ambiguous and appears inconsistent with the preceding equality −KEz = −nE|Ez if 'quadric' is allowed to mean a quadric cone. On an n-dimensional quadric cone Q with hyperplane class H, one has −KQ = (n−1)H and H·ℓ = 1, whereas −nE|Ez would have degree n on the ruling line ℓ. Thus the equality forces a smooth quadric, or at least rules out the vertex cone. The terminology should be clarified and the proof adjusted accordingly.

minor comments (4)

[Lemma 3.1] The step 'Fixing p ∈ Ez a general point ... we have dim Uz,p ≥ n+1' is not justified as written: if p is not in the image of the evaluation map, the fibre over p is empty. The argument should either choose p in the image of ev or explain how dominance of ev is obtained.
[Section 2] The reference [Kol96, Prop. II.1.3] in Lemma 3.1 and [Kol96, Theorem II.1.3] in Lemma 3.2 should be checked and cited consistently, since the exact hypotheses of the cited statement matter for the main argument.
[Lemma 5.1] The smoothness verification of X is relegated to 'a straightforward calculation' without showing the Jacobian or its rank. Since the example in Section 5 is meant to illustrate the main result, it would be helpful to include the computation or a reference.
[Throughout] There are typographical issues such as 'eiher' for 'either' in Lemma 3.1 and 'K ebekus' in the abstract; these should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main dichotomy and classification branches are derived from external deformation-theoretic and Mori-theoretic inputs, not from the theorem being assumed.

full rationale

The paper's derivation chain contains no fitted parameters, no prediction that is its own input, and no load-bearing self-citation. The dichotomy -E·Γ ∈ {1,2} in Proposition 3.3 follows from the submaximal length hypothesis l(f)=n-1, adjunction on the general fibre, and external deformation-family dimension bounds from Kol96 together with the external characterizations of projective space and quadrics in CMSB02, Ke02, DH17, and Mi04. The projective-bundle and quadric-bundle conclusions in Theorem 4.1 and Theorem 4.5 are built from established relative adjunction and evaluation-map results (AW93, BS95, Hö12, Kol11, FG65), not from a restatement of the target theorem. Section 5 provides an independent concrete example rather than a disguised assumption. A possible reviewer concern about the applicability of Kol96 Proposition II.1.3 to singular fibres would be a correctness risk about an external cited bound, not circularity: the paper does not define the target result in terms of that bound, and no self-citation chain forces the dichotomy. No circular step was found.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The theorem is proved entirely on the basis of external established results in Mori theory and deformation theory; no unknown constants are fitted and no new objects are postulated.

assumptions (8)

standard math Ionescu-Wisniewski inequality (Theorem 1.3): dim E + dim F >= dim X + l(f) - 1.
Used to define maximal and submaximal length; the setup fixes l(f) = dim E_z - 1.
standard math [CMSB02, Theorem 0.1]: an unsplit family of rational curves of minimal degree in a Fano fibre has dimension at most 2n-2, with equality iff the fibre is normalized by P^n.
Invoked in Prop. 3.3 and Lemma 4.2; it yields the -E.Gamma in {1,2} dichotomy.
standard math [Deb01, Theorem 7.39.c]: relative numerical triviality (K_X - nE) equiv_f 0 implies relative linear equivalence (K_X - nE) sim_f 0 under the Fano assumption.
Used in Lemma 3.1 and Lemma 4.4 to turn degree computations into divisor relations.
standard math [Kol96, II.1.3]: dimension estimate for deformation families of rational curves, dim H >= -K_X.Gamma - 3 + dim X, and bend-and-break.
Used in Lemmas 3.1, 3.2, and Prop. 3.3 to produce large families of minimal curves.
standard math [Kol96, I.3.12]: constancy of line bundle degree on members of a closed deformation family over a normal base.
Used in Lemma 2.3 to show families are unsplit and all members are contracted by f.
standard math [Ro68, Thm 3.5] and [AW93, Thm 5.1]: existence of a birational model with a vector bundle and surjective evaluation along fibres.
Used in Lemma 2.8 and Thm 4.5 to construct the quadric bundle model.
standard math [BS95, Thm 3.1.6] (Kobayashi-Ochiai type): a normal n-fold with -K_X = nL for an ample L is a quadric or P^n.
Used in Prop. 3.3 to identify irreducible fibres in the -E.Gamma=1 case as quadrics.
standard math [Kol11, Thm 12] (simultaneous normalization) and [FG65] (local analytic triviality of families of projective spaces).
Used in Lemmas 4.2 and 4.3 to globalize the fibrewise P^n conclusion.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Divisorial Mori contractions of submaximal length." pith.science (2026). https://pith.science/paper/GBI36LML

@misc{pith2026241117549,
  author       = {Pith},
  title        = {Pith review of: Divisorial Mori contractions of submaximal length},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GBI36LML}},
  note         = {Machine review of arXiv:2411.17549}
}

read the original abstract

A result due to Cho, Miyaoka, Shepherd-Barron [CMSB] and Kebekus [Ke] provides a numerical characterization of projective spaces. More recently, Dedieu and H\"oring [DH] gave a characterization of smooth quadrics based on similar arguments. As a relative version of [CMSB] and [Ke], H\"oring and Novelli proved in [HN] that the locus covered by positive-dimensional fibres in a Mori contraction of maximal length is a projective bundle up to birational modification. We change the length hypothesis and we prove that the exceptional locus of a divisorial Mori contraction of submaximal length is birational either to a projective bundle, or to a quadric bundle.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Wiśniewski, A note on nonvanishing and applications

Marco Andreatta and Jaroslaw A. Wiśniewski, A note on nonvanishing and applications. Duke Mathematical Journal 72, No. 3, 1993

work page 1993
[2]

Beltrametti and Andrew J

Mauro C. Beltrametti and Andrew J. Sommese, The adjunction theory of complex projective varieties. Volume 16 of de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin, 1995

work page 1995
[3]

Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds

Koji Cho, Yoichi Miyaoka and Nicholas I. Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds. Advanced Studies in Pure Mathematics 35, p. 1--88. Mathematical Society of Japan, 2002

work page 2002
[4]

Universitext, Springer, 2001

Olivier Debarre, Higher-dimensional algebraic geometry. Universitext, Springer, 2001

work page 2001
[5]

Algebraic geometry 4, No

Thomas Dedieu and Andreas Höring, Numerical characterization of quadrics. Algebraic geometry 4, No. 1, p. 120--135, 2017

work page 2017
[6]

Nachrichten der Akademie der Wissenschaften in Göttingen 2, Vandenhoeck & Ruprecht, 1965

Wolfgang Fischer and Hans Grauert, Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten. Nachrichten der Akademie der Wissenschaften in Göttingen 2, Vandenhoeck & Ruprecht, 1965

work page 1965
[7]

Graduate Texts in Mathematics, Springer, 1977

Robin Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, Springer, 1977

work page 1977
[8]

Andreas Höring and Carla Novelli, Mori contractions of maximal length. Publ. RIMS 49, No. 1, p. 215--228, 2013

work page 2013

Show all 18 references

[9]

Journal of Algebraic Geometry 21, p

Andreas Höring, On a conjecture by Beltrametti and Sommese. Journal of Algebraic Geometry 21, p. 721--751, 2012

2012
[10]

Mathematical Proceedings of the Cambridge Philosophical Society 99, p

Paltin Ionescu, Generalized adjunction and applications. Mathematical Proceedings of the Cambridge Philosophical Society 99, p. 457--472, 1986

1986
[11]

In Complex geometry (Göttingen, 2000), p

Stefan Kebekus, Characterizing the projective space after Cho, Miyaoka and Shepherd-Barron. In Complex geometry (Göttingen, 2000), p. 147-–155. Springer, Berlin, 2002

2000
[12]

Volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete

János Kollár, Rational curves on Algebraic Varieties. Volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, 1996

1996
[13]

Asian Journal Of Mathematics 15, No

János Kollár, Simultaneous normalization and algebra husks. Asian Journal Of Mathematics 15, No. 3, p. 437--450, 2011

2011
[14]

With the collaboration of Sándor Kovács

János Kollár, Singularities of the minimal model program. With the collaboration of Sándor Kovács. Cambridge University Press, 2013

2013
[15]

In Complex analysis in several variables --- Memorial conference of Kiyoshi Oka's centennal birthday

Yoichi Miyaoka, Numerical characterisations of hyperquadrics. In Complex analysis in several variables --- Memorial conference of Kiyoshi Oka's centennal birthday. Advanced Studies in Pure Mathematics 42, p. 209--235, Mathematical Society of Japan, 2004

2004
[16]

Hugo Rossi, Picard variety of an isolated singular point, Rice University Studies 54, No. 4, p. 63–-73, 1968

1968
[17]

Wiśniewski, On contractions of extremal rays of Fano manifolds

Jaroslaw A. Wiśniewski, On contractions of extremal rays of Fano manifolds. Journal für die reine und angewandte Mathematik 417, p. 141--158, 1991

1991
[18]

Transactions of the American Mathematical Society 53, p

Oscar Zariski, Foundations of a general theory of birational correspondences. Transactions of the American Mathematical Society 53, p. 490--542, 1943

1943

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Wiśniewski, A note on nonvanishing and applications

Marco Andreatta and Jaroslaw A. Wiśniewski, A note on nonvanishing and applications. Duke Mathematical Journal 72, No. 3, 1993

work page 1993

[2] [2]

Beltrametti and Andrew J

Mauro C. Beltrametti and Andrew J. Sommese, The adjunction theory of complex projective varieties. Volume 16 of de Gruyter Expositions in Mathematics. Walter de Gruyter & Co., Berlin, 1995

work page 1995

[3] [3]

Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds

Koji Cho, Yoichi Miyaoka and Nicholas I. Shepherd-Barron, Characterizations of projective space and applications to complex symplectic manifolds. Advanced Studies in Pure Mathematics 35, p. 1--88. Mathematical Society of Japan, 2002

work page 2002

[4] [4]

Universitext, Springer, 2001

Olivier Debarre, Higher-dimensional algebraic geometry. Universitext, Springer, 2001

work page 2001

[5] [5]

Algebraic geometry 4, No

Thomas Dedieu and Andreas Höring, Numerical characterization of quadrics. Algebraic geometry 4, No. 1, p. 120--135, 2017

work page 2017

[6] [6]

Nachrichten der Akademie der Wissenschaften in Göttingen 2, Vandenhoeck & Ruprecht, 1965

Wolfgang Fischer and Hans Grauert, Lokal-triviale Familien kompakter komplexer Mannigfaltigkeiten. Nachrichten der Akademie der Wissenschaften in Göttingen 2, Vandenhoeck & Ruprecht, 1965

work page 1965

[7] [7]

Graduate Texts in Mathematics, Springer, 1977

Robin Hartshorne, Algebraic Geometry. Graduate Texts in Mathematics, Springer, 1977

work page 1977

[8] [8]

Andreas Höring and Carla Novelli, Mori contractions of maximal length. Publ. RIMS 49, No. 1, p. 215--228, 2013

work page 2013