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On the Nef cones of blowups of the projective plane

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Pith's one-line read The Cremona action on the nef cone of the blowup of the projective plane in nine very general points admits a rational polyhedral fundamental cone with ten facets and ten rays.

desk verdict A promising and mostly checkable geometric result—explicit rational polyhedral fundamental cones for the n=9 nef cone and the n≥10 K-negative nef effective cone—but the n≥10 part currently rests on an unstated lemma and a corrupted Section 6, so it needs careful revision before I'd trust it. read the letter →

arxiv 2412.15460 v2 pith:2DWUSU62 submitted 2024-12-19 math.AG math.GR

Luíze D'Urso This is my paper · ORCID
classification math.AGmath.GR MSC 14E0714J2614C2020F5551F15
keywords nefconeCremonaactionfundamentaldomainblowupoftheprojectiveplanehyperbolicreflectiongroupCoxetersimplexverygeneralpointsconjecture
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 studies the Cremona action—the group generated by quadratic plane transformations centered at triples of blown-up points—on the cone of numerically effective divisors of the blowup of the projective plane in n very general points. It proves that for n=9 this nef cone, although it has infinitely many extremal rays, admits a rational polyhedral fundamental cone with exactly 10 facets and 10 extremal rays. For every n≥10, it proves the same for the part of the nef effective cone on which the canonical class is non-positive, with a fundamental cone having n+2 facets. The argument goes through hyperbolic geometry: the nef cone is cut by the hyperboloid H^n, the Cremona action becomes a reflection group, and the fundamental domain turns out to be a Coxeter polytope. The nine-point polytope is one of only three hyperbolic Coxeter 9-simplices, so the description cannot be simplified.

What carries the argument

The central object is the pair (W, H^n): the Neron-Severi space of the blowup is the Minkowski space $R^{{1,n}}$, the Cremona group W is generated by the reflections w0 (the standard Cremona centered at P1, P2, P3) and the coordinate permutations, and W preserves the hyperboloid H^n. The load-bearing mechanism is the reflection-group criterion quoted as Theorem 2.15: a Coxeter polytope—one whose facet angles are submultiples of π—is a fundamental domain for the reflection group it generates. The polytope P_n = {x0 ≥ −x1−x2−x3, x1 ≤ ⋯ ≤ xn ≤ 0} is proved to be a fundamental domain for β_n, and the combinatorial Claim 4.6 is the engine that shows every −1-curve inequality is a sum of the basic degree-0 and degree-1 inequalities. For n=9, containment in the light cone is checked directly and the boundary has exactly two rays, yielding a finite-volume simplex; for n≥10, the extra halfspace 3x0 ≥ −Σxi coming from the K-negative condition restores finiteness.

What would settle it

Find a nef class on the blowup of $P^{2}$ in nine very general points with $D^{2}$=0 that is not W-equivalent to either (1,−1,0,…,0) or (3,−1,…,−1); Theorem 5.2 would fail, and the same would happen if a nef effective class with K·D≤0 on an n≥10 blowup were found that is not W-equivalent to any point satisfying the defining inequalities of C_n.

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Extended reading notes

Core claim

The central claim, stated as Theorems 1.3 and 1.6, is that the cone C_n cut out by x0 ≥ −x1−x2−x3, x1 ≤ x2 ≤ ⋯ ≤ xn ≤ 0, and for n≥10 also 3x0 ≥ −Σxi, is a fundamental domain: every point of the relevant nef cone is W-equivalent to a point of C_n, and no two interior points of C_n are equivalent. For n=9 this is a fundamental domain of the full nef cone of the blowup of $P^{2}$ in nine very general points, and C_9 is the convex cone over a hyperbolic Coxeter 9-simplex with 10 vertices corresponding to: a line, a line through P1, a conic through P1 and P2, and cubics through P1,…,Pi for i=3,…,9. For n≥10 the same explicit cone is a fundamental domain for the K-negative part Nefe(S_n)_{K≤0}, the classes in the nef effective cone that pair non-positively with the canonical class. The proof identifies the projectivized nef cone with a subset β of H^n, shows the polytope P=tilde P∩β is cut by the listed inequalities using a decomposition of every −1-curve inequality into line and conic inequalities, and then verifies finite volume by showing the cone lies in the light cone, with boundary intersection consisting of the W-orbits of e0−e1 and −K.

Load-bearing premise

The argument relies on a quoted identification, not proved in this paper, that the K-negative part of the nef effective cone is exactly the intersection of the −1-curve halfspaces, and for n=9 that the only nef classes on the boundary of the light cone are the W-orbits of e0−e1 and −K.

Editorial extensions

If this is right

  • For n=9, the non-polyhedral, infinitely generated nef cone is completely encoded by the action of the infinite Cremona group on a 10-ray simplex: every nef class can be brought into C_9 by a Cremona move.
  • The nine-point fundamental cone has the minimum possible number of facets and rays for a full-dimensional cone in R^10 not containing a line, so the description is optimal in that sense.
  • Because P_9 is a Coxeter simplex, reflecting it across its facets tiles all of H^9; adding the reflection in x9=0 gives a discrete reflection group for which P_9 is a fundamental domain.
  • For n≥10, every nef effective class with K·D≤0 is Cremona-equivalent to a point of an explicit cone with n+2 facets, and for n=10, 11, 13 the corresponding hyperbolic polytope is again Coxeter.
  • For n=9 the nef classes on the boundary of the light cone are exactly the W-orbits of e0−e1 and −K, so C_9 together with these two orbits describes all nef classes.

Reading between the lines

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

  • The paper leaves open whether the same explicit cone remains a fundamental domain for special configurations of nine points, such as base points of a pencil of cubics, where the automorphism group is infinite; the very-generality assumption may be stronger than the Coxeter-simplex mechanics requires.
  • If the −1-curves conjecture is eventually proved for all n, the K≤0 restriction in Theorem 1.6 could become unnecessary, because the current proof needs that restriction to force finite volume of the intersection with H^n.
  • The method suggests a broader principle: for varieties with trivial or small automorphism groups but large groups of birational transformations preserving the canonical class, the cone conjecture should be formulated for that larger group rather than for Aut alone.
  • The fact that the fundamental polytope is Coxeter exactly for n=10, 11, 13 indicates that the tiling property is rare and dimension-dependent; one could ask whether the nef cones in those dimensions carry extra symmetries inherited from the reflection group.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper studies the Cremona action on the nef cone of the blowup Sn of P^2 in n very general points, for n ≥ 9. The author defines a cone Cn cut out by explicit inequalities (x0 ≥ −x1−x2−x3, x1 ≤ x2 ≤ ... ≤ xn ≤ 0, and, for n ≥ 10, 3x0 ≥ −Σxi) and claims that Cn is a rational polyhedral fundamental domain: for n = 9 for the whole nef cone (Theorem 1.3), and for n ≥ 10 for the K-negative part of the nef effective cone (Theorem 1.6). The proof passes through hyperbolic geometry: the Cremona group is realized as a reflection group on H^n, the nef cone is related to the intersection of halfspaces defined by −1-curves, and the candidate fundamental polytope Pn = Cn ∩ H^n is shown to have finite volume in the relevant cases. The paper also proves that the n = 9 polytope is one of the three hyperbolic Coxeter 9-simplices (Theorem 1.5).

Significance. If the central claims hold, the paper gives a concrete, explicit rational polyhedral fundamental domain for the Cremona action on the nef cone in the n = 9 case and on the K-negative nef effective cone for n ≥ 10, connecting a birational geometry problem to the classification of hyperbolic Coxeter simplices. The paper's internal combinatorial arguments, such as the induction in Claim 4.6 and the Cartan matrix computations, are explicit and checkable. However, the proofs of Theorems 1.3 and 1.6 rely on structural statements about the nef cone that are quoted from the literature but not stated precisely or proved in the paper, and Section 6 contains corrupted passages that prevent verification of Proposition 6.1 and Theorem 6.2.

major comments (4)
  1. [§3, Remark 3.9 and Definition 3.12] The identification of the K-negative part of Nef_e(Sn) with the cone β ∩ {v·K_S ≤ 0}, where β is cut out by all −1-curve halfspaces, is load-bearing for Theorem 6.2. This identification is made via the assertion NE(S) ⊂ R(−K_S) + Σ R_i, attributed to [dF10, Lemma 4.1], but the lemma is not stated, its hypotheses are not given, and its range of n is not specified. If the lemma is not valid for all n ≥ 10 in the required form, then Theorem 6.2 only constructs a fundamental domain for a cone that may be strictly larger than Nef_e(Sn)_{K≤0}. The author should either state and prove the needed version of the lemma or give a precise statement with citation and explain exactly how it applies for general n.
  2. [§5, Theorem 5.2] The proof of Theorem 5.2 uses the assertion that 'the nef elements in ∂L9 are spanned by the elements in the orbit W(1,−1,0,...,0) and −K_S' without proof or precise citation. This boundary-ray description is essential: it is exactly what converts the statement that C ∩ L^○_9 is a fundamental domain for Nef(S) ∩ L^○_9 into the global statement that ∪_{w∈W} wC = Nef(S). The description is recalled in §3 without a reference to a specific theorem or proof. The author should either prove this fact or give a precise citation to a statement that covers the very general 9-point blowup.
  3. [§3, Remark 3.11 and §6, Theorem 6.2 proof] Theorem 6.2's boundary argument only treats rational rays on ∂L^n ('Let v = (x0,...,xn) ∈ R^{1,n} be a point in a rational ray of Nef(S) ∩ ∂Ln such that v·K_S ≤ 0'). The paper's own Remark 3.11 states that Nef(S)/Nef_e(S) ⊂ ∂L is the union of the irrational rays, and this remark is used implicitly to conclude that checking rational boundary rays suffices for covering Nef_e(S)_{K≤0} ∩ ∂L. The remark is stated as 'well known' but not proved, and it is not clear that it applies to the very general blowups under consideration. This is a second unstated input in the boundary-coverage argument; it should be stated explicitly and either proved or precisely cited.
  4. [§6, Proposition 6.1 and Theorem 6.2] The text of Section 6 is corrupted by embedded non-text sequences (for example, '⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪⟨⟨⟪rl⟫mo⟨⌟⟪⟨⟨⟪rl⟫mo...' and '9 coordinates−1'), which appear inside the statements of Proposition 6.1 and the proof of Theorem 6.2. These corruptions make the claimed vertices, the boundary-ray enumeration, and the final covering argument impossible to verify. The author must supply a clean version of the full Section 6 before the proof can be assessed.
minor comments (6)
  1. [§3, first paragraph] The text contains a typo: 'in general posistion' should be 'in general position'.
  2. [Remark 1.4] The phrase 'the the convex cone' should read 'the convex cone'.
  3. [§5, proof of Theorem 5.2] The proof contains the stray symbol string 'Leftr⫯g⊸tl⫯ne⇒', which appears to be a formatting artifact; it should be removed or replaced with the intended implication sign.
  4. [§5, after Cartan matrix computation] The sentence 'besides being a fundamental domain of βn, we can obtain Theorem 2' refers to 'Theorem 1.5'; the numbering should be corrected.
  5. [References] The reference [dF10] is listed as 'On the mori cone of blow-ups of the plane' with the date '01 2010'; the paper should provide the full bibliographic data, including journal or preprint series, so that the quoted Lemma 4.1 can be located.
  6. [§2, Definition 2.2 and later] There is some notational inconsistency between writing vectors as (x0, x1, ..., xn) and points in projective space as (x0 : x1 : ... : xn); the paper should explicitly state that scalar multiples are identified in H^n throughout, especially when discussing boundary points in §5 and §6.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the fundamental-domain construction is a self-contained Coxeter/reflection-group argument, and the geometric inputs (fef10, n=9 boundary description) are prior external results, not the paper's own conclusions.

full rationale

The derivation chain is not circular. The central construction proceeds by defining the cone C_n by explicit inequalities, proving in Theorem 4.5 (with the self-contained Claim 4.6) that every -1-curve halfspace is redundant, and then applying Lemma 4.4, which is a general and trivially proved statement about intersections of fundamental domains with invariant subsets. Propositions 5.1 and 6.1 give direct computations showing that the resulting polytopes have finite volume and identifying their boundary points. The only load-bearing facts imported from outside are: the Coxeter polytope theorem (Theorem 2.15, cited to Dolgachev), the classification of hyperbolic Coxeter simplices (cited to AVS93), the known n=9 description of the nef cone boundary as W-orbits of e0-e1 and -K_S, and [dF10, Lemma 4.1] used in Remark 3.9 to identify the K-negative nef cone with the -1-curve halfspace cone. None of these is a result of this paper, none is a self-citation, and none is equivalent to the theorems being proved; they are genuine external inputs. There are no fitted parameters, no quantity is fitted to a subset of data and then called a prediction, and no target claim is assumed inside the proof. The n>=10 result is honestly stated for the cone cut out by -1-curve inequalities plus K<=0, with the identification to Nef_e(S)_{K<=0} explicitly resting on the cited [dF10, Lemma 4.1]; whether that lemma is strong enough is a correctness/rigor concern, not a circularity concern. Overall the paper is self-contained in its combinatorial heart and non-circular in its use of external geometry.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No new particles or entities are introduced. The constants such as sqrt(2) and sqrt(2/(n-9)) are normalization choices for normal vectors, not free parameters fitted to data.

assumptions (4)
  • standard math Theorem 2.15 of Dolgachev: every Coxeter polytope in H^n is a fundamental domain for the reflection group generated by its facets.
    Cited as [Dol07], used in the proof of Theorem 4.1 to assert that the polytope defined by the chosen halfspaces is a fundamental domain.
  • domain assumption For n=9, Conjecture 3.6 holds and the boundary rays of the nef cone are the W-orbits of e0, e0-e1 and -K_S.
    Used in Section 3 and Theorem 5.2 to conclude that the W-orbit of the explicit cone covers the whole nef cone. The paper states this as known but gives no precise theorem citation at the point of use.
  • domain assumption Lemma 4.1 of [dF10]: NE(S) is contained in R(-K_S) plus the sum of the -1-curve rays.
    Used in Remark 3.9 and Definition 3.12 to identify the K-negative part of the nef cone with the intersection of -1-curve halfspaces. This is external to the paper and load-bearing for n at least 10.
  • standard math Classification of hyperbolic Coxeter simplices in [AVS93]: exactly three exist in H^9 and none for n greater than 9.
    Used for Theorem 1.5 and Proposition 6.3 to match the n=9 simplex to the classification and to note that higher-dimensional cases cannot be simplices.

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Pith. "Pith review of On the Nef cones of blowups of the projective plane." pith.science (2026). https://pith.science/paper/2DWUSU62

@misc{pith2026241215460,
  author       = {Pith},
  title        = {Pith review of: On the Nef cones of blowups of the projective plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2DWUSU62}},
  note         = {Machine review of arXiv:2412.15460}
}
abstract

In this paper, we study the Cremona action on the nef cone of $S_n$, the blowup of $\P^2$ in $n$ very general points, with $n\ge9$. We construct and describe a rational polyhedral fundamental domain of the nef cone for $n=9$ with respect to this action. In the case $n\ge10$, we give a rational polyhedral fundamental domain of the $K_{S_n}$-negative part of the nef cone with respect to the Cremona action.

Figures

Figures reproduced from arXiv: 2412.15460 by the authors.

Figure 1
Figure 1. A sketch of Nef(S) for 3 ≤ n ≤ 8. We are actually interested in the case n ≥ 9, when there are infinitely many −1-curves, and each of them generates an extremal ray, so NE(S) and Nef(S) are not polyhedral. When n = 9, the extremal rays of NE(S) are gen￾erated by the infinitely many −1-curves, which form an orbit of the action of W, and also the ray spanned by −KS, which is a limit ray. Similarly, the ex￾tremal rays … view at source ↗
Figure 2
Figure 2. A sketch of Nef(S) for n = 9. When n ≥ 10, Conjecture 3.6 is not known to hold. What we do know is that if {Ri}i∈I are all the rays spanned by −1-curves ci , then the Mori cone NE(S) contains the cone generated by the light cone and all Ri , while [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. A sketch of Nef(S) for n = 10 if conjecture 3.6 holds. Definition 3.10. The nef effective cone of S is the intersection Nefe (S) = Nef(S) ∩ Eff(S). Remark 3.11. It is well known that Nef(S) ⊂ L ⊂ NE(S). In fact, Nef(S)/Nefe (S) ⊂ ∂L is the union of the irrational rays of Nef(S) ∩ ∂L . For n ≤ 9, Nef(Sn) = Nefe (Sn). Definition 3.12. The KS-negative side of Nefe (S) is Nefe (S)KS≤0 ∶ = {v ∈ Nefe (S) ∣ v ⋅ KS ≤ 0} = {… view at source ↗

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Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Blowups, Gale duality, and moduli spaces

    math.AG 2026-05 unverdicted novelty 7.0 of 10

    The K-negative part of the effective cone of the blowup of P^n at n+4 general points is generated by the Weyl-group orbit of one exceptional divisor, and the movable cone decomposes into Mori chambers.

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