{"id":"e9c08d75-d973-4303-9eca-5302b3f20a57","arxiv_id":"2506.14118","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New infinite families of affine Calabi-Yau manifolds with irregular toric tangent cones are constructed, with an explicit algorithm for the Reeb field and Minkowski decompositions.","lead":"The authors construct new families of complete Calabi-Yau spaces that stretch to infinity like irregular cones, answering an open question about how many such examples exist. They also provide working computer code for computing the Reeb field and all maximal Minkowski decompositions of a toric Calabi-Yau cone.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 5.5's exclusion of the rational candidate a=0 rests on an inequality with the wrong sign; the r=1 irregularity claim of Theorem A is not proven as written, though the smoothing and metric existence parts are unaffected.","rationale":"I agree with the reader's conditional verdict. The main structural results, especially the existence of affine smoothings and AC Calabi--Yau metrics on them, are supported by explicit Minkowski decompositions and known deformation/gluing theorems, and I do not see a comparable gap there. The weakest point is precisely the irregularity proof for the r=1 Cho--Futaki--Ono family. The inequality in Proposition 5.5 is false as written: the cubic is decreasing on the relevant interval, so the direction of the inequality is reversed. The numerical minimizers and the claimed irreducibility of the degree-8 polynomial suggest the conclusion is likely repairable, but as written the proof does not exclude the rational value a=0. This is a proof gap, not a demonstrated false conclusion, so the appropriate outcome is a conditional acceptance pending a corrected or independently verified argument. The reader's weakest_assumption identified the same Proposition 5.5 gap, and I focus on it rather than on the black-box computation issue because the polynomial is explicit and can be independently checked once the a=0 exclusion is settled.","tokens_in":43636,"tokens_out":7156,"duration_ms":82102,"concrete_test":"Run an exact decision-procedure check (e.g., Mathematica Reduce with exact rational arithmetic, or a Gröbner-basis elimination in Singular) on the system ∂_a a0 = 0, ∂_b a0 = 0, a=0, c=3, together with the five Reeb-cone inequalities displayed in Proposition 5.5, for s=2,3,4 and symbolically for all s≥2. If the system is inconsistent, replace the false monotonicity sentence with a certified exclusion of a=0 and Proposition 5.5 is repaired; if it is consistent, the r=1 irregularity assertion in Theorem A is not established by the given argument and the paper's main novelty would need reassessment.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the new smoothings are AC Calabi--Yau with irregular tangent cones has two parts. The smoothing and metric existence part is well supported by the explicit maximal Minkowski decomposition in Theorem 5.1, Altmann's versal-base theorem, and [21, Theorem 4.3]. The load-bearing weak point is the proof of irregularity for the r=1 Cho--Futaki--Ono family in Proposition 5.5. To rule out a=0, the proof substitutes a=0 into the equation ∂_b a0=0 and obtains the cubic factor f(b)=b^3+9b^2+b(27-18s^3)+27(s^4-2s^3+1). It then asserts that since b<3(s-1), one has f(b)<27s^3(1-s)<0. This is not valid: on the relevant interval f'(b)=3(b+3)^2-18s^3, which is negative for s≥2, so f is decreasing and b<3(s-1) implies f(b)>f(3(s-1)), not the reverse. For s=3 the cubic already has roots in the admissible interval (-3,6). Thus the rational candidate a=0 is not excluded. Since the remaining factor P(a) is claimed irreducible of degree 8, excluding a=0 is the only step preventing a rational first coordinate of the minimizer. The reliance on Mathematica's Minimize and IrreduciblePolynomialQ adds a secondary reproducibility risk, but the false inequality is the concrete mathematical gap.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs new complete asymptotically conical (AC) Calabi–Yau metrics on affine smoothings of irregular toric Calabi–Yau cones. Theorem A treats the Cho–Futaki–Ono family: each cone is shown to admit an r-parameter affine smoothing over an irreducible versal base, with AC Calabi–Yau metrics on the smooth fibers, and the r=1 members are claimed to be irregular. Theorem B treats Minkowski sums of Gauntlett–Martelli–Sparks–Waldram polygons with a lattice segment, giving two families of one-parameter smoothings with AC Calabi–Yau metrics. The paper also provides Macaulay2 and Mathematica code for computing Hilbert series, Minkowski decompositions, and Reeb minimizers, and it benchmarks these computations against the known del Pezzo cone examples.","tokens_in":43878,"tokens_out":26820,"duration_ms":253611,"significance":"If the main claims are correct, this gives the first infinite families of affine Calabi–Yau manifolds with irregular tangent cones after the original example of Conlon–Hein, thereby answering a question from [21]. The paper's computational toolkit is a genuine strength: the code is included, the algorithms are explicit, and the known del Pezzo cone values are reproduced. The deformation-theoretic framework via Altmann's versal-base theorem is well matched to the problem. However, the irregularity assertion for the r=1 subfamily rests on a specific inequality in Proposition 5.5 that is false as written, so the strongest advertised conclusion is not currently established.","major_comments":[{"comment":"The proof of irregularity for P_s contains an invalid inequality that is load-bearing for the r=1 case of Theorem A. After substituting a=0 into the equation ∂_b a_0=0, the paper obtains the cubic f(b)=b^3+9b^2+b(27-18s^3)+27(s^4-2s^3+1) and asserts that b<3(s-1) implies f(b)<27s^3(1-s)<0. This is backwards: f'(b)=3(b+3)^2-18s^3 is negative on the relevant interval for s≥2, so f is decreasing and b<3(s-1) implies f(b)>f(3(s-1))=27s^3(1-s). Moreover, for s=3 the cubic has a root in the admissible interval (-3,6), since f(0)=756>0 and f(6)=-1458<0. Thus the rational candidate a=0 is not excluded by the argument given, and the conclusion that the minimizer's first coordinate must be a root of the irreducible degree-8 polynomial P is not justified. The smoothing and metric-existence parts of Theorem A survive, but the irregularity claim for the Cho–Futaki–Ono r=1 cones is not proven as written.","section":"Section 5.2, Proposition 5.5"},{"comment":"The displayed maximal Minkowski decomposition in Theorem 5.1 is inconsistent with the printed examples. For s=3, P_3 is defined as Conv((-1,-2),(0,-1),(1,3),(0,2),(-1,-1)), which after translating by (1,2) has vertices (0,0),(1,1),(2,5),(1,4),(0,3). But with L_1=Conv((0,0),(1,1)) and Δ_3=Conv((0,0),(0,1),(1,4)) as stated, the Minkowski sum L_1+Δ_3 has convex hull (0,0),(1,1),(1,2),(2,5),(1,4),(0,1), a hexagon rather than P_3. Consequently the theorem's statement of Δ_s (or the vertex list of P_s) needs correction before Theorem 5.1 can be used to justify the smoothing construction for all r,s.","section":"Section 5.1 and Example 5.7"},{"comment":"The proof of generic smoothness is incomplete as written. The argument invokes Hartshorne III, Lemma 10.5 after establishing that π_0 is dominant, but that lemma normally requires a nonsingular source (or a separate argument that the singular locus does not dominate the base) and it yields an open subset of the base over which the map is smooth, not an open subset of X_0 as stated in the lemma. Since the existence of at least one smooth fiber is needed for the application of [21, Theorem 4.3] in Corollary 5.3 and in the proof of Theorem B(b), this step should be justified directly, for example by a Jacobian computation on the explicit toric total space or by a precise citation from Altmann's theory.","section":"Lemma 5.2"}],"minor_comments":[{"comment":"The sentence \"Then Y_3 is the intersection of the following 20 quadrics in C^10\" appears to contain a typo: the displayed weight matrix W has nine columns, so the ambient space should be C^9.","section":"Example 3.9"},{"comment":"The displayed list of quadrics for the ideal I contains an isolated double comma in the line \"z_4 z_6 - z_1 z_9, z_5^2 - z_1 z_9, , z_4 z_5 - z_1 z_8\"; this is a harmless typo but should be cleaned up.","section":"Example 3.8"},{"comment":"The code relies on Mathematica's Minimize and IrreduciblePolynomialQ as black boxes. A short note on the exactness of these calls, or a certificate such as a minimal polynomial and a root-isolation interval, would make the irregularity verification reproducible without trusting undocumented internals.","section":"Appendix A.3"},{"comment":"The notation Q_{p,q} for the Minkowski sums in Theorem B conflicts with the earlier notation Q_i for del Pezzo toric diagrams; this is manageable but occasionally confusing.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper delivers the first affine Calabi-Yau smoothings for the Cho-Futaki-Ono cones and a clean classification for the GMSW-plus-segment family, and that is real progress. But the proof that the r=1 subfamily is irregular rests on an inequality with the wrong sign, so that specific claim is unproven as written.\n\nThe structural parts are the strong parts. Theorem 5.1 identifies the unique maximal Minkowski decomposition for the Cho-Futaki-Ono family, and Theorem B gives a complete answer for when Y^{p,q}+L is smoothable. The authors use Altmann's deformation theory carefully, and the volume-minimization pipeline is genuinely useful, with good benchmarks against known del Pezzo cones in Examples 3.8-3.10. The code is included, though not in a versioned repository.\n\nThe concrete flaw is in Proposition 5.5. To exclude the rational candidate a=0, they substitute into the partial derivative and obtain a cubic f(b). They claim that b<3(s-1) implies f(b)<27s^3(1-s)<0. The inequality has the sign backwards: f'(b)=3(b+3)^2-18s^3 is negative for s≥2, so f is decreasing on the relevant interval, and b<3(s-1) implies f(b)>f(3(s-1))=27s^3(1-s). Thus the rational a=0 is not excluded. Since the remaining degree-8 factor is claimed irreducible, this is the only step preventing a rational Reeb coordinate, so the irregularity conclusion for r=1 is not established. The claim may still be true; the numerics and Example C are suggestive. But the proof needs repair.\n\nSecondary concerns: the irreducibility check and the global minimization rely on Mathematica's IrreduciblePolynomialQ and Minimize, which is fine in principle but hard to audit without a pinned script. Also, the heavy use of [21] is self-citation, but those cited results are external theorems, not fitted parameters, so I do not see a circularity problem.\n\nImportantly, the smoothing and AC Calabi-Yau metric existence parts of Theorem A do not depend on the flawed inequality. They follow from the Minkowski decomposition plus [21, Theorem 4.3], and those steps look sound. So this is not a broken paper; it is a valuable contribution with a fixable gap.\n\nWho should read it: people working on AC Calabi-Yau metrics, toric deformations, and Sasakian geometry will get new examples and a reusable computational method. I would send it to a serious referee, and I would ask the authors to fix Proposition 5.5 or supply an alternative argument for irrationality before accepting.","headline":"Solid new smoothing examples and a useful Minkowski-decomposition pipeline, but the proof that the r=1 Cho-Futaki-Ono cones are irregular has a load-bearing sign error.","tokens_in":44497,"tokens_out":2848,"would_cite":true,"duration_ms":27698,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["53C25","53C55","14M25","14B07"],"pacs":[],"model":"deepseek-v4-flash","headline":"Every toric Calabi–Yau cone in the $P_{r,s}$ family admits an $r$-parameter affine smoothing carrying asymptotically conical Calabi–Yau metrics, and for $r=1$ the tangent cone is irregular; adding suitable lattice segments to $Y^{p,q}$…","keywords":["Calabi–Yau metrics","asymptotically conical manifolds","toric Calabi–Yau cones","irregular Reeb field","Minkowski decomposition","versal deformation","affine smoothing","volume minimization"],"falsifier":"Take $s=2$ and $a=0$, and compute the cubic $b^3+9b^2+b(27-18s^3)+27(s^4-2s^3+1)$ at values of $b$ in the allowed interval $(-3,3(s-1))$, comparing with the claimed bound $27s^3(1-s)$; if the cubic is not always negative, the exclusion of $a=0$ fails and the irregularity proof needs repair. One can also test directly whether $a=0$ solves the volume-minimizer equations, or check whether the degree-8 polynomial $P(a)$ has a rational root in $(-3,3)$.","tokens_in":43362,"feed_emoji":"📐","tokens_out":8158,"duration_ms":77000,"temperature":0.7,"pith_summary":"This paper constructs new complete Ricci-flat Kähler (Calabi–Yau) metrics on affine algebraic manifolds whose tangent cone at infinity is an irregular toric Calabi–Yau cone, for a whole family of cones rather than a single known example. The main result is that every cone associated with the family of lattice polygons $P_{r,s}$ (with $2r+3$ vertices and $s-1$ interior lattice points) admits an $r$-parameter affine smoothing over an irreducible base $\\mathbb{C}^r$, and each smooth fiber carries an asymptotically conical Calabi–Yau metric with the original cone as its tangent cone; for $r=1$ the cone is irregular. The paper also gives a complete criterion for when adding a lattice segment to a quadrilateral toric diagram $Y^{p,q}$ produces a new toric diagram with a lattice maximal Minkowski decomposition, yielding one-parameter smoothings with Calabi–Yau metrics. Along the way it provides explicit computer code to compute the Reeb field (the volume minimizer) and all Minkowski decompositions of any toric Calabi–Yau cone with smooth link.","feed_headline":"Irregular cones are smoothed into new Calabi–Yau manifolds","feed_subtitle":"Each cone in a large toric family gets an r-parameter affine smoothing with Ricci-flat metrics.","key_machinery":"The machinery has three main parts. First, a toric diagram is a convex lattice polygon with no interior lattice points on its edges, encoding a three-dimensional toric Gorenstein (hence Calabi–Yau) cone. Second, deformation theory identifies the versal base of the cone with the Minkowski scheme built from the polygon's lattice maximal Minkowski decompositions: decompositions $P=P_0+\\cdots+P_m$ into lattice segments and triangles correspond to $m$-parameter smoothings over $\\mathbb{C}^m$. Third, the Calabi–Yau Reeb field is the unique minimizer of the normalized volume function, computed from the index character via Hilbert series, and the cone is irregular exactly when a coordinate of this minimizer is irrational. The paper's contribution is to wire these three parts into an algorithm and to execute it on the $P_{r,s}$ and $Y^{p,q}+L$ families.","core_discovery":"The central claim is that irregularity of the tangent cone is not an obstruction to affine smoothing: each toric Calabi–Yau cone $C_{r,s}$ in the $P_{r,s}$ family is the tangent cone at infinity of an asymptotically conical Calabi–Yau manifold obtained by an $r$-parameter affine smoothing, and for $r=1$ the Calabi–Yau Reeb field has an irrational coordinate, making the cone irregular. The proof identifies the versal deformation base with the reduced Minkowski scheme $\\mathbb{C}^r$ via the unique lattice maximal Minkowski decomposition $P_{r,s}=L_1+\\cdots+L_r+\\Delta_s$, applies a general existence theorem for AC Calabi–Yau metrics on smooth fibers of such deformations, and computes the Reeb field as the unique minimizer of the normalized volume. For the quadrilateral family $Y^{p,q}$, the paper proves that $Y^{p,q}+L$ has a lattice maximal Minkowski decomposition exactly for the two listed choices of segment $L$, and that the resulting cone is the tangent cone of an AC Calabi–Yau metric on a one-parameter smoothing. A conjecture is formulated: $Y^{p,q}+L$ is irregular exactly when $Y^{p,q}$ is.","pith_inferences":["If the paper's conjecture holds, then the two segment choices produce irregular cones for every irregular $Y^{p,q}$, so the construction would yield an infinite family of irregular tangent cones with smoothings beyond the $r=1$ subfamily.","The same pipeline could be run on any lattice polygon with a maximal Minkowski decomposition, giving an explicit though case-by-case method to decide smoothability and irregularity; the main limitation is computational rather than conceptual.","Because the existence of AC Calabi–Yau metrics on the smoothings does not use the irregularity assertion, the smoothing results in Theorems A and B are independent of the correctness of the inequality in Proposition 5.5; only the 'irregular' label depends on that step.","An exact-arithmetic re-run of the minimization for small $s$ could decide the $a=0$ exclusion directly, and would either repair or refute the claimed irrationality of the Reeb field in Proposition 5.5."],"forward_implications":["Every cone $C_{r,s}$ admits an $r$-parameter affine smoothing, and every smooth fiber carries a complete Calabi–Yau metric with Euclidean volume growth and quadratic curvature decay asymptotic to the cone.","For $r=1$ these smoothings provide infinitely many new affine Calabi–Yau manifolds with irregular tangent cone, answering a question left open by the classification of asymptotically conical Calabi–Yau manifolds.","For the two allowed segments $L$, the cones obtained from $Y^{p,q}+L$ are one-parameter smoothable and appear as tangent cones of AC Calabi–Yau metrics on the smoothing.","Combined with known crepant resolutions, the smoothings give geometric transitions from the resolved to the deformed Calabi–Yau metrics through the cone.","The supplied code computes the Reeb field and all Minkowski decompositions from the toric diagram alone, making the irregularity test and the deformation-theoretic smoothability check algorithmically accessible."],"supporting_citations":[{"why":"Supplies the general existence theorem for AC Calabi–Yau metrics on smooth fibers of affine deformations, used in every smoothing statement.","marker":"[21]"},{"why":"Provides the versal deformation theory identifying the Minkowski scheme with the versal base of a toric Gorenstein singularity.","marker":"[2]"},{"why":"Introduces the $P_{r,s}$ family of toric Calabi–Yau cones and the equivalence between toric Gorenstein cones and toric diagrams.","marker":"[14]"},{"why":"Defines the quadrilateral family $Y^{p,q}$ of irregular toric Calabi–Yau cones that Theorem B modifies by adding a lattice segment.","marker":"[28]"},{"why":"Gives the theorem that a toric Kähler cone admits a Calabi–Yau metric exactly when its Reeb field minimizes the volume function.","marker":"[27]"},{"why":"Establishes the volume-minimization framework and index character used to compute the Reeb field explicitly.","marker":"[42]"},{"why":"Provides the first example of an affine Calabi–Yau manifold with irregular tangent cone, the baseline that the new families extend.","marker":"[20]"},{"why":"Gives the rigorous algebraic treatment of the index character and Hilbert series that underlies the computer algorithm.","marker":"[18]"}],"fun_headline_variants":["Irregular toric cones obtain new Calabi-Yau smoothings","New Calabi-Yau manifolds from irregular cone smoothings","Smoothing irregular cones yields Ricci-flat Calabi-Yau metrics","Explicit Reeb fields give Calabi-Yau metrics on irregular cones","Minkowski decompositions produce Calabi-Yau smoothings of irregular cones"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper's irregularity conclusion for the $r=1$ cones rests on the computer-assisted claim that the volume-minimizing Reeb field has an irrational coordinate, together with an inequality in Proposition 5.5 that is stated incorrectly as written; if the minimization or the exclusion of the rational candidate $a=0$ fails, the irregularity assertion would need revision, although the existence of Calabi–Yau metrics on the smoothings would survive.","fun_headline_variants_meta":{"raw":{"variants":["Irregular toric cones obtain new Calabi-Yau smoothings","New Calabi-Yau manifolds from irregular cone smoothings","Smoothing irregular cones yields Ricci-flat Calabi-Yau metrics","Explicit Reeb fields give Calabi-Yau metrics on irregular cones","Minkowski decompositions produce Calabi-Yau smoothings of irregular cones"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000189,"raw_usage":{"total_tokens":1341,"prompt_tokens":953,"completion_tokens":388,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":569,"completion_tokens_details":{"reasoning_tokens":308}},"tokens_in":569,"tokens_out":388,"duration_ms":4367,"temperature":1.0,"reasoning_tokens":308,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:18:49.398348+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $s=2$ and $a=0$, and compute the cubic $b^3+9b^2+b(27-18s^3)+27(s^4-2s^3+1)$ at values of $b$ in the allowed interval $(-3,3(s-1))$, comparing with the claimed bound $27s^3(1-s)$; if the cubic is not always negative, the exclusion of $a=0$ fails and the irregularity proof needs repair. One can also test directly whether $a=0$ solves the volume-minimizer equations, or check whether the degree-8 polynomial $P(a)$ has a rational root in $(-3,3)$.","supporting_citations":[{"cited_title":"J.173(2024), no","cited_arxiv_id":null,"evidence_quote":"Supplies the general existence theorem for AC Calabi–Yau metrics on smooth fibers of affine deformations, used in every smoothing statement."},{"cited_title":"Math.128(1997), no","cited_arxiv_id":null,"evidence_quote":"Provides the versal deformation theory identifying the Minkowski scheme with the versal base of a toric Gorenstein singularity."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the $P_{r,s}$ family of toric Calabi–Yau cones and the equivalence between toric Gorenstein cones and toric diagrams."},{"cited_title":"Gauntlett, D","cited_arxiv_id":null,"evidence_quote":"Defines the quadrilateral family $Y^{p,q}$ of irregular toric Calabi–Yau cones that Theorem B modifies by adding a lattice segment."},{"cited_title":"Futaki, H","cited_arxiv_id":null,"evidence_quote":"Gives the theorem that a toric Kähler cone admits a Calabi–Yau metric exactly when its Reeb field minimizes the volume function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the volume-minimization framework and index character used to compute the Reeb field explicitly."},{"cited_title":"Conlon and H.-J","cited_arxiv_id":null,"evidence_quote":"Provides the first example of an affine Calabi–Yau manifold with irregular tangent cone, the baseline that the new families extend."},{"cited_title":"Collins and G","cited_arxiv_id":null,"evidence_quote":"Gives the rigorous algebraic treatment of the index character and Hilbert series that underlies the computer algorithm."}],"review_version":1}