{"id":"761c9686-4115-43d8-88d6-ce27a54eaf7a","arxiv_id":"2412.15460","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For blowups of the projective plane at 9 very general points, the nef cone has a 10-inequality fundamental domain under Cremona transformations; for n at least 10, the K-negative part of the nef effective cone has one too.","lead":"This paper finds a finite wedge-shaped piece of the nef cone for blowups of the plane at 9 very general points, so that every cone point can be moved into this piece by a Cremona transformation. For 10 or more points, the same construction works for the part of the cone where the canonical class is negative.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The n≥10 result is proved for the -1-curve halfspace cone, and the identification of that cone with Nef_e(S_n)_{K≤0} rests entirely on an unstated form of [dF10, Lemma 4.1]; if that lemma is weaker than needed, Theorem 1.6 is not established.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing point: the equality of the K-negative nef effective cone with the intersection of -1-curve halfspaces, via [dF10, Lemma 4.1] and the n=9 boundary-ray description. My reading of Sections 3-6 confirms that Theorem 6.2 constructs a fundamental domain for the -1-curve halfspace cone, and the passage from that cone to Nef_e(S_n)_{K≤0} is made by quoting an external lemma rather than stating it. This concern is about verifiability and completeness of the proof, not about a demonstrated mathematical falsehood. The explicit cone C_n is plausible, and the internal Coxeter-group argument for the halfspace cone appears coherent apart from presentation issues in Section 6. Therefore the appropriate verdict remains conditional: the paper should state the precise form of [dF10, Lemma 4.1], verify that it applies for all n≥10, and clean up Section 6 so Proposition 6.1 and Theorem 6.2 can be checked. I do not see grounds to reject the central claim, but I also do not see the n≥10 theorem as fully established in the current text.","tokens_in":25650,"tokens_out":39900,"duration_ms":334281,"concrete_test":"Read [dF10, Lemma 4.1] and verify verbatim whether it states NE(S_n) ⊂ R_{\\ge0}(−K_{S_n}) + \\sum_i R_{\\ge0} c_i for every n≥9 very general blowup of P^2. Then re-derive Remark 3.9 by dualizing: the claimed equality Nef(S_n)∩{K≤0} = {v·K≤0, v·c_i≥0 ∀i} follows only if the lemma has no extra hypotheses and the inclusion is in the stated direction. If the lemma is weaker, supplies only the opposite inclusion, or is restricted to n=9, then Theorem 6.2 proves a fundamental domain for a different cone and Theorem 1.6 is unproved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim for n≥10 is Theorem 1.6. The proof in Section 6 constructs a fundamental domain for β∩{K≤0}, where β=⋂_{−1-curve c}(√2 c)≥0 (Definition 4.2). The identification of β∩{K≤0} with Nef_e(S_n)_{K≤0} is made in Remark 3.9 and Definition 3.12 through the assertion NE(S)⊂R(−K_S)+∑R_i, quoted from [dF10, Lemma 4.1]. No statement of the lemma, its hypotheses, or the range of n is given. This is load-bearing: if [dF10, Lemma 4.1] is weaker than stated, or is only valid for n=9, then Theorem 6.2 proves a fundamental domain for a cone that is only known to contain Nef_e(S_n)_{K≤0}, and the reverse inclusion is exactly what makes the Cremona orbits cover the geometric nef cone. The n=9 theorem similarly depends on the recalled boundary description in Theorem 5.2 (the W-orbits of e0−e1 and −K_S), which is asserted without a precise citation. A second unstated input is Remark 3.11, which claims that non-effective nef rays are precisely the irrational rays on ∂L; Theorem 6.2 only treats rational boundary rays, so boundary coverage of Nef_e relies on this unproved remark as well. These are external results, but the paper does not state them, leaving the central claim not self-contained at its most critical point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":22,"tokens_out":3018,"duration_ms":448192,"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":[{"comment":"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.","section":"§3, Remark 3.9 and Definition 3.12"},{"comment":"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.","section":"§5, Theorem 5.2"},{"comment":"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.","section":"§3, Remark 3.11 and §6, Theorem 6.2 proof"},{"comment":"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.","section":"§6, Proposition 6.1 and Theorem 6.2"}],"minor_comments":[{"comment":"The text contains a typo: 'in general posistion' should be 'in general position'.","section":"§3, first paragraph"},{"comment":"The phrase 'the the convex cone' should read 'the convex cone'.","section":"Remark 1.4"},{"comment":"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.","section":"§5, proof of Theorem 5.2"},{"comment":"The sentence 'besides being a fundamental domain of βn, we can obtain Theorem 2' refers to 'Theorem 1.5'; the numbering should be corrected.","section":"§5, after Cartan matrix computation"},{"comment":"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.","section":"References"},{"comment":"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.","section":"§2, Definition 2.2 and later"}],"recommendation":"major_revision","confidential_remarks":"The paper's central idea is attractive, but the main theorems depend on external results that are not stated precisely. In particular, the author should be asked to state the version of [dF10, Lemma 4.1] used and confirm that it covers all n ≥ 10. The corruption in Section 6 is severe and must be fixed before any further evaluation. The n = 9 part may be salvageable with a precise citation for the boundary-ray description, but as written the proof is not self-contained at its critical points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague, quick take on arXiv:2412.15460. The core new content is real: for S_9, the nef cone's Cremona action gets an explicit rational polyhedral fundamental cone with 10 facets, and for n≥10 the K-negative part Nef_e(S_n)_{K≤0} gets one with n+2 facets. The construction via the Coxeter polytope P_n = P̃_n ∩ β_n and the induction reducing -1-curve halfspaces to the three listed inequalities is clean. Claim 4.6 and Proposition 5.1 appear checkable and likely correct. The identification of P_9 as one of the three hyperbolic Coxeter 9-simplices is a nice bonus. So the architecture is sound and the main theorems are new; I don't see circularity or parameter-fitting.\n\nThe soft spots are real but localized. Section 6 is badly corrupted—there is literal garbage text in Proposition 6.1, its proof, the vertex list, and Theorem 6.2. As it stands those statements cannot be checked, and that is the section proving Theorem 1.6. That alone requires a full rewrite before the paper is usable.\n\nThe second issue is that the link between the -1-curve halfspace cone β and the actual nef cone is imported through [dF10, Lemma 4.1] with no statement of the lemma or its hypotheses. The equality Nef_e(S_n)_{K≤0} = β∩{K≤0}∩Eff is load-bearing for n≥10, and the proof of Theorem 6.2 also relies on the claim that non-effective nef rays are precisely the irrational boundary rays (Remark 3.11) and on the n=9 boundary ray description in Section 3, both asserted without proof or precise citation. These may be correct and known, but the paper needs to state them explicitly; as written the central theorem is not self-contained at its most critical point.\n\nIf those external inputs are as strong as the author needs, Theorems 1.3 and 1.6 are established. The likely fix is a revision: clean Section 6, quote the de Fernex lemma properly, and give a precise reference for the n=9 nef boundary. The n=9 part is in decent shape; the n≥10 part is conditional.\n\nI would send this to a serious referee. The idea is good, the result is the kind of concrete cone-conjecture step people will use, and the flaws are fixable rather than fatal. I'd not cite it until the Section 6 text is restored and the external lemmas are stated.","headline":"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.","tokens_in":26551,"tokens_out":4161,"would_cite":false,"duration_ms":37850,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E07","14J26","14C20","20F55","51F15"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["nef cone","Cremona action","fundamental domain","blowup of the projective plane","hyperbolic reflection group","Coxeter simplex","very general points","cone conjecture"],"falsifier":"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.","tokens_in":25435,"feed_emoji":"📐","tokens_out":11352,"duration_ms":72127,"temperature":0.7,"pith_summary":"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.","feed_headline":"A 10-ray cone captures every nef class of the nine-point blowup","feed_subtitle":"Cremona moves reduce every nef class on the nine-point blowup to one explicit simplex; the K-negative part works for all n≥10 too.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Theorem 2.15, the criterion that a Coxeter polytope is a fundamental domain for the reflection group it generates; this is the bridge from geometry to the group action.","marker":"[Dol07]"},{"why":"Supplies Lemma 4.1, used to identify the K-negative part of the nef cone with the intersection of the −1-curve halfspaces in Theorems 5.2 and 6.2.","marker":"[dF10]"},{"why":"Gives the classification table of hyperbolic Coxeter simplices used to identify the nine-point polytope as one of the three Coxeter 9-simplices.","marker":"[A VS93]"},{"why":"Frames the −1-curves conjecture for very general blowups of the plane, whose n=9 case is used for the nef cone description.","marker":"[dF05]"}],"fun_headline_variants":["10-ray cone pins down every nef class on nine-point blowup","Nef cone of nine-point blowup: one explicit simplex captures all","A single cone absorbs Cremona moves for blowups of P^2","Cremona action tamed: one cone for n=9 and K≤0 for n≥10"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["10-ray cone pins down every nef class on nine-point blowup","Nef cone of nine-point blowup: one explicit simplex captures all","A single cone absorbs Cremona moves for blowups of P^2","Cremona action tamed: one cone for n=9 and K≤0 for n≥10"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000195,"raw_usage":{"total_tokens":1365,"prompt_tokens":962,"completion_tokens":403,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":578,"completion_tokens_details":{"reasoning_tokens":317}},"tokens_in":578,"tokens_out":403,"duration_ms":4688,"temperature":1.0,"reasoning_tokens":317,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T11:25:25.766639+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}