{"id":"6ed6808c-2738-4d22-afce-9b2dd81a8b58","arxiv_id":"2605.27152","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"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.","lead":"This paper shows that the blowup of n-dimensional projective space at n+4 general points has a K-negative effective cone generated by a single Weyl-group orbit, with a described chamber decomposition. It identifies the blowup with a moduli space of rank-two sheaves on a Gale-dual plane blowup.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Gale duality is assumed to preserve Cremona-generality (Notation 2.7) without proof; if a Cremona-general P can have a Gale dual Q that is not Cremona-general, the wall classification and Theorem 6.1 lack a necessary hypothesis.","rationale":"The reader's weakest assumption correctly identifies the Gale-duality preservation of Cremona-generality as the point where the logical chain is least secure. The paper's main theorem is stated for any Cremona-general P, but the proof needs the Gale dual Q to be Cremona-general in order to classify the K-negative stability walls and to prove Lemma 4.13, which is then transported to X by the determinant map. Without a proof or citation that Cremona-generality is invariant under Gale duality, there is a real gap: the theorem may hold, but the argument does not cover all configurations satisfying its hypothesis. I considered the alternative concern that Theorems 5.7 and 6.1 are only proved for n>3, leaving n=2,3 uncovered; this is a genuine coverage issue, but it is secondary because those cases have finite Weyl group and are likely accessible by known Mori-dream-space results, whereas the Gale-duality issue affects the infinite-Weyl-group cases that are the paper's main novelty. The rest of the argument is substantial and appears internally coherent, so I do not see grounds for rejecting the paper; the correct response is to keep the reader's conditional verdict pending a justification of Notation 2.7 or an explicit strengthening of the theorem's hypothesis.","tokens_in":42985,"tokens_out":29316,"duration_ms":271334,"concrete_test":"Use the matrix description of Gale duality in Remark 2.4. For a Gale dual pair (P,Q), let I⊂{1,...,n+4} with |I|=n+1, and let J be the 3-subset corresponding to I under the Dynkin isomorphism of Notation 2.7 (β_0↦α_0, β_i↦α_{k−i}). Compute the Gale dual of the Cremona-transformed configuration I^*P and compare it with J^*Q. If they are projectively equivalent for every I, then the transformation rule holds for all finite sequences and Notation 2.7 is justified; if the equality fails for a single I, exhibit that as a counterexample. A finite symbolic check for n=3,4 (where W is finite) plus a general matrix computation for arbitrary n would settle whether the preservation statement is true.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Notation 2.7 assumes without proof that if the configuration P⊂P^n is Cremona-general, then its Gale dual Q⊂P^2 is also Cremona-general. This is load-bearing: Section 4's wall classification (Prop. 4.3, Lemma 4.10, Lemma 4.13) and hence Theorems 6.1 and 1.7 are proved for S=Bl_Q P^2 with Q Cremona-general. In particular, Lemma 4.13 identifies E with the (closed) cone generated by W·(h−e_1), and this classification of numerical rational classes uses the Cremona-generality of Q. If some Cremona-general P has a Gale dual Q that, after a finite sequence of plane Cremona transformations, contains three collinear points, then the identities Nef(S)_{K≤0}=... and E=sum_{C∈W·(h−e_1)} R_{\\ge0}C are not justified, and the determinant map ρ might not send E onto Eff(X)_{K≤0}. The proof neither proves nor cites the needed preservation statement, and the main theorem's hypothesis is 'Cremona-general', not 'very general', so a generic density argument cannot be invoked without an additional argument. This is a condition on the input data, not on the Mori-theoretic claim itself, but it is required for the proof as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the birational geometry of X = Bl_{n+4} P^n, the blowup of P^n at n+4 Cremona-general points. Following Mukai, it uses Gale duality to identify X with a Gieseker moduli space of rank-2 torsion-free sheaves on S = Bl_{n+4} P^2, the blowup of P^2 at the Gale-dual points. The paper then analyzes the variation of these moduli spaces as the polarization on S varies, introduces a determinant map rho: N^1(S) -> N^1(X), and claims that rho transfers the stability chamber decomposition of Nef(S)_{K<=0} to the Mori chamber decomposition of Eff(X)_{K<=0}. The main theorem (Theorem 1.7) asserts that Eff(X)_{K<=0} is the cone generated by the Weyl-group orbit of E_1, and that Mov(X)_{K<=0} is the union of pullbacks of nef cones of countably many small modifications of X.","tokens_in":43350,"tokens_out":29482,"duration_ms":289594,"significance":"If the main theorem is correct, it gives a complete description of the K-negative part of the pseudoeffective and movable cones for a substantial family of non-Mori-dream-space blowups, and it answers Questions 1.6 in the case k=n+4. The paper contains substantial technical work: a detailed wall-crossing analysis for moduli spaces on blowups of P^2, an explicit determinant map, and a careful use of the Coble pairing and Weyl-group action. The claimed link between moduli wall-crossing and the birational geometry of X is an attractive and potentially influential idea. However, several load-bearing points are not adequately proved, and one boundary case appears to contradict the stated main theorem.","major_comments":[{"comment":"The paper assumes, without proof, that if P is a Cremona-general configuration in P^n, then its Gale dual Q in P^2 is also Cremona-general. This assumption is used immediately: Section 4 (Proposition 4.3, Lemma 4.13) and hence Theorem 6.1 require S = Bl_Q P^2 to be blown up at Cremona-general points. But Theorem 1.7 only assumes P is Cremona-general. The proof never proves or cites preservation of Cremona-generality under Gale duality. This is not a mere technicality: the wall classification of K-negative walls and the identification of E with the cone over W.(h-e_1) can fail if Q possesses three collinear points after a Cremona sequence. A density argument for 'very general' points would need an explicit statement that the Gale dual of a very general configuration is very general; the paper's hypothesis is 'Cremona-general', not merely 'very general', and the needed implication is absen","section":"Notation 2.7; Sections 4 and 6"},{"comment":"Theorem 5.7 constructs the explicit determinant map rho only under the assumption n > 3. Yet Theorem 1.7 is stated for all n >= 2. The later proof of Theorem 1.7 through Theorem 6.1 relies on this explicit rho. No separate treatment is given for n = 2 and n = 3. For n = 3 (k=7) and n = 2 (k=6) the varieties are Mori dream spaces and the cone statements may be recoverable by finite polyhedral arguments, but that is not written. As it stands, the main theorem is not proved for the full range claimed.","section":"Theorem 5.7 vs. Theorem 1.7"},{"comment":"There is an apparent contradiction in the k=9 case, which corresponds to n=5. Lemma 4.13(1) asserts E = E', where E' is the cone generated by W.(h-e_1). But for k=9, equation (14) gives -K_S in E: -K_S is nef, K_S.(-K_S)=0, e.(-K_S)=1 for e in W.e_1, and (2B+K_S).(-K_S)=6 for B in W.h. On the other hand, -K_S is not in E', because every generator C in W.(h-e_1) has C.(-K_S)=2, while (-K_S)^2=0, so any nonnegative combination of such C has nonnegative intersection with -K_S, and equality forces all coefficients to be zero; hence -K_S cannot be a positive combination. Thus E != E'. Remark 4.14 effectively acknowledges this by adding an R_{>=0}(-K) ray, and Corollary 6.2 adds a corresponding R_{>=0}(-K_X) ray. But Theorem 6.1(3), used in Theorem 1.7, still states the equality without that ray. For X = Bl_9 P^5, -K_X = 6H - 4 sum E_i is not effective by a dimension count (a degree-6 hypersur","section":"Lemma 4.13, Remark 4.14, Corollary 6.2"}],"minor_comments":[{"comment":"In the proof of Theorem 5.7, the text says 'l, f_1, ..., f_n form a Z-basis for N_1(X)'. Since X = Bl_{n+4}P^n has Picard rank n+5, the basis should be l, f_1, ..., f_{n+4}.","section":"Proof of Theorem 5.7"},{"comment":"The displayed line defining E' reads 'E' := ... subset E' subset Nef(S)_{K<=0}', which is self-referential. It should presumably be 'E' := ... subset E subset Nef(S)_{K<=0}'.","section":"Section 4.3"},{"comment":"The paper uses 'Cremona-general' in the main theorem but 'very general position' in the abstract and in the opening sentence. These are different notions in general, and the distinction matters for the Gale-duality hypothesis; the wording should be aligned.","section":"Throughout"}],"recommendation":"reject","confidential_remarks":"The k=9/n=5 issue seems to be a genuine false statement in the main theorem, not just a missing proof. If the authors can restrict the main theorem to n != 5 and prove the exceptional case separately, the paper might become salvageable, but as submitted the central claim is not correct as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read. This is a substantial step on Questions 1.6 for k=n+4. The main theorem — describing Eff(X)_{K≤0} as the cone spanned by the W-orbit of E_1 and giving a chamber decomposition via moduli wall-crossing — is genuinely new for arbitrary n, and the determinant-map mechanism is a real tool, not a repackaging. The moduli-side work (Sections 3–4) is careful: the wall classification for K-negative walls, the identification of the special chamber C_0 with the blowup, and the description of Π and E are coherent and detailed. The proof of Theorem 6.1 is a plausible chain: ρ maps supporting hyperplanes of chambers to supporting hyperplanes of nef cones, E to Eff_{K≤0}, etc. If the technical hypotheses hold, the argument goes through.\n\nBut the technical hypotheses are where I have problems.\n\nFirst, Notation 2.7 simply assumes that the Gale dual of a Cremona-general configuration is Cremona-general. That is not proved or cited. It is load-bearing: Section 4's wall classification and Lemma 4.13 require S to be blown up at Cremona-general points. If a Cremona-general P can have a Gale dual Q with three collinear points after some Cremona sequence, the identities E=E' and Nef(S)_{K≤0}=... fail, and ρ might not map E onto Eff(X)_{K≤0}. This is a condition on the input data, but it's needed for the proof as written. It may be true — I suspect it is — but it needs an argument.\n\nSecond, Theorem 5.7 constructs ρ only for n>3, but Theorem 1.7 claims n≥2. For n=2 and 3, the statement may well follow from known results or a limiting argument, but the paper doesn't say. Theorem 6.1 uses ρ without restating the n>3 assumption, and the proof of Theorem 1.7 does not supply the missing cases. This is an actual gap between the claim and the proof.\n\nNeither gap looks fatal. The architecture is sound and the gaps are addressable. But they are not cosmetic. A referee should ask for them to be fixed before publication.\n\nWho is this for? Specialists in birational geometry and moduli of sheaves. It deserves a serious referee. My recommendation: send it to review, but with a request to fill these two holes and to be explicit about the range of n in the main theorems.","headline":"Strong paper on the K-negative birational geometry of Bl_{n+4}P^n, but the proof as written has two gaps: Gale duality preserving Cremona-generality is assumed without proof, and the determinant map is only constructed for n>3 while the theorem claims n≥2.","tokens_in":43815,"tokens_out":3194,"would_cite":true,"duration_ms":32402,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14E07","14J60","14D20","14E30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that for X = Bl_{n+4}P^n at Cremona-general points, the K-negative pseudoeffective cone is generated by the Weyl orbit of one exceptional divisor, and the K-negative movable cone is a union of nef cones of small modificatio","keywords":["Gale duality","blowups of projective space","moduli spaces of sheaves","Mori chamber decomposition","pseudoeffective cone","Weyl group","Cremona transformations","Coble pairing"],"falsifier":"Perform the finite classification of divisor classes $D$ on $\\mathrm{Bl}_{10}\\mathbb{P}^6$ with $K\\cdot D < 0$ using the bounds given in Proposition 4.7. The theorem predicts that every such class is a nonnegative combination of the Weyl orbit $W\\cdot E_1$; finding a single $D$ with $D^2 > 0$ outside that cone would disprove Theorem 1.7(1).","tokens_in":42897,"feed_emoji":"📐","tokens_out":10073,"duration_ms":100048,"temperature":0.7,"texified_at":"2026-08-05T21:06:39.185978+00:00","pith_summary":"The paper aims to describe the birational geometry of the blowup of $\\mathbb{P}^n$ at $n+4$ points in a sufficiently general (Cremona-general) position. Its central theorem states that, on the side of the canonical class that is negative, the effective cone is spanned by a single Weyl-group orbit of exceptional divisors, and the movable cone is a countable union of nef cones of small modifications. This matters because $n+4$ is the smallest number of points for which such blowups admit infinitely many rational contractions, so the theorem gives a complete structural description of that infinite complexity. The proof works by realizing the blowup as a Gieseker moduli space of rank-2 sheaves on a Gale-dual blowup of the plane and translating polarization wall-crossings into birational contractions.","texify_model":"deepseek-v4-flash","texify_usage":{"total_tokens":7326,"prompt_tokens":865,"completion_tokens":6461,"prompt_tokens_details":{"cached_tokens":0},"prompt_cache_hit_tokens":0,"prompt_cache_miss_tokens":865,"completion_tokens_details":{"reasoning_tokens":5569}},"feed_headline":"One Weyl orbit generates the K-negative cone of P^n blowups","feed_subtitle":"For Bl_{n+4}P^n, every birational contraction on the negative side of the canonical class comes from a Gale-dual moduli wall.","key_machinery":"Gale duality for $n+4$ points in $\\mathbb{P}^n$ and $\\mathbb{P}^2$, together with the Weyl group $W_{n,n+4}$ acting by reflections through a $(-2)$-root lattice in the Picard group (the Coble pairing: $(H,H)=n-1$, $(H,E_i)=0$, $(E_i,E_j)=-\\delta_{ij}$). The paper's engine is the moduli-space realization $X \\cong M_{C_0}$, where $M_{C_0}$ is the Gieseker moduli space of $L$-semistable rank-2 torsion-free sheaves on $S$ with $c_1 = -K_S$ and $c_2 = 2$, for a polarization $L$ in a specific chamber. Variation of polarization across $K$-negative walls gives explicit blowups and flips of moduli spaces, described by non-split extensions $0 \\to \\mathcal{O}(D) \\to E \\to \\mathcal{O}(-K-D) \\to 0$; the determinant map $\\rho$ transfers this wall-and-chamber structure to the birational geometry of $X$.","core_discovery":"The paper's central claim is Theorem 1.7: for $X = \\mathrm{Bl}_{n+4}\\mathbb{P}^n$ with Cremona-general points, $\\mathrm{Eff}(X)_{K\\le 0}$ is the cone spanned by the Weyl group orbit of one exceptional divisor, and $\\mathrm{Mov}(X)_{K\\le 0}$ is the union of nef cones of countably many smooth small modifications $g_i: X \\dashrightarrow Y_i$, with each $Y_i$ having polyhedral nef cone away from the $K=0$ boundary. This is established by realizing $X$ as a Gieseker moduli space $M_{C_0}$ of rank-2 torsion-free sheaves ($c_1 = -K_S, c_2 = 2$) on the Gale-dual blowup $S$ of $\\mathbb{P}^2$, and then using the stability wall-and-chamber decomposition of the ample cone of $S$ to describe the cones. The determinant map $\\rho: N^1(S) \\to N^1(X)$ converts every stability wall into a supporting hyp","pith_inferences":["If Gale duality preserves Cremona-generality — the paper's unproved input — the same moduli-space strategy should extend to other values of s, realizing Bl_{n+s+2}P^n as a moduli space of sheaves on Bl_{n+s+2}P^s; the paper leaves this as an expectation (Remark 2.6), but the mechanism here makes it a testable program.","The determinant-map dictionary suggests a general principle: for blowups with a dual configuration, birational contractions should be indexed by destabilizing sub-line-bundles on the dual surface; this could yield practical algorithms for computing Mori chambers for other point counts, such as k = n+5.","A finite computational check for n=6 (k=10) — enumerating all numerical divisor classes with K·D<0 using the bounds of Proposition 4.7 and testing membership in the Weyl orbit cone — would verify Theorem 1.7(1) in the smallest untreated case."],"forward_implications":["Every extremal ray of Eff(Bl_{n+4}P^n) in the K≤0 half-space is accounted for by the Weyl orbit of one exceptional divisor, so all K-negative contractions fit a single combinatorial pattern.","The movable cone on the K≤0 side is a countable union of nef cones, each corresponding to a small modification X ⇢ Y_i, giving a partial Mori chamber decomposition for these non-Mori-dream-space blowups.","For n=5, k=9, the formula becomes Eff(X) = R_{\\ge0}(−K_X) + ∑_{E∈W·E_1} R_{\\ge0}E, a closed description of the full effective cone (Corollary 6.2).","The moduli realization gives a constructive way to produce every K-negative rational contraction: it is induced by a wall-crossing of Gieseker stability on the Gale-dual surface.","This answers Questions 1.6 affirmatively for k = n+4, providing the first full description of K-negative birational geometry for the minimal number of points with infinitely many contractions."],"fun_headline_variants":["Gale duality turns blowups into moduli spaces","K-negative cone of blowups is a single Weyl orbit","Blowups at n+4 points are Gieseker moduli spaces","Mori chambers from Gale-dual stability walls","Weyl orbit spans the negative side of Eff"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The paper assumes, without proof, that Gale duality sends a Cremona-general configuration of $n+4$ points in $\\mathbb{P}^n$ to a Cremona-general configuration in $\\mathbb{P}^2$; all the cone statements in Sections 4 and 6 depend on the dual surface being a blowup of $\\mathbb{P}^2$ at Cremona-general points.","fun_headline_variants_meta":{"raw":{"variants":["Gale duality turns blowups into moduli spaces","K-negative cone of blowups is a single Weyl orbit","Blowups at n+4 points are Gieseker moduli spaces","Mori chambers from Gale-dual stability walls","Weyl orbit spans the negative side of Eff"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000232,"raw_usage":{"total_tokens":1341,"prompt_tokens":775,"completion_tokens":566,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":519,"completion_tokens_details":{"reasoning_tokens":484}},"tokens_in":519,"tokens_out":566,"duration_ms":5987,"temperature":1.0,"reasoning_tokens":484,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-02T12:59:55.633502+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Perform the finite classification of divisor classes $D$ on $\\mathrm{Bl}_{10}\\mathbb{P}^6$ with $K\\cdot D < 0$ using the bounds given in Proposition 4.7. The theorem predicts that every such class is a nonnegative combination of the Weyl orbit $W\\cdot E_1$; finding a single $D$ with $D^2 > 0$ outside that cone would disprove Theorem 1.7(1).","supporting_citations":[],"review_version":2}