Toric Landau-Ginzburg models in threefold divisorial contractions

Artan Sheshmani; Yang He

arxiv: 2605.20126 · v1 · pith:C762K3HOnew · submitted 2026-05-19 · 🧮 math.AG

Toric Landau-Ginzburg models in threefold divisorial contractions

Yang He , Artan Sheshmani This is my paper

Pith reviewed 2026-05-20 03:20 UTC · model grok-4.3

classification 🧮 math.AG

keywords toric Landau-Ginzburg modelsdivisorial contractionsquantum periodsFano threefoldsterminal singularitiesmirror symmetrySarkisov linksregularized periods

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The pith

For certain divisorial contractions of terminal Fano threefolds the regularized quantum period of the base equals the infinite-r limit of the period on the exceptional divisor.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that when g: Y to X is a divisorial contraction between Q-factorial terminal Fano threefolds and the center is a smooth point, terminal quotient point, cA/n point or smooth curve with cA or cA/n singularities, the regularized periods satisfy lim r to infinity of Ĝ_Y,rE(t) equals Ĝ_X(t). The proof works inside the framework of toric Landau-Ginzburg models. A reader cares because the identity supplies an explicit mirror-symmetric way to move quantum-period data across the contraction. This in turn yields a computational route to Sarkisov links and higher syzygies of the central models.

Core claim

Assuming the center of the contraction g:Y to X is either a smooth point, a terminal quotient point, a point of type cA/n, or a smooth curve with singularities of type cA or cA/n, the regularized period identity lim r to +infinity Ĝ_Y,rE(t) equals Ĝ_X(t) holds, where Ĝ denotes the regularized quantum period of the toric Landau-Ginzburg model attached to the pair.

What carries the argument

The regularized quantum period Ĝ of a toric Landau-Ginzburg model attached to the pair (variety, divisor), with the limit taken as the coefficient r of the exceptional divisor tends to infinity.

If this is right

Quantum periods of the base X can be recovered from those of the blow-up Y once the limit is taken.
Sarkisov links between threefolds become computable via the period relation.
Higher syzygies of central models in dimension three acquire a mirror-symmetric description.
The identity supplies a recursive way to move period data down a sequence of contractions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same limiting relation might be tested on other classes of terminal singularities not covered by the four listed types.
If the identity survives deformation, it could give a period-based obstruction to the existence of certain links.
Numerical evaluation of the periods for a concrete cA/n contraction would supply an immediate consistency check.
The construction may extend to non-terminal or non-Q-factorial cases once suitable regularizations are defined.

Load-bearing premise

The contraction center must be one of the four listed types: smooth point, terminal quotient point, cA/n point, or smooth curve with cA or cA/n singularities.

What would settle it

An explicit example of one of the allowed contraction types in which the numerical value of the limit of Ĝ_Y,rE(t) differs from Ĝ_X(t) would falsify the claimed identity.

read the original abstract

We investigate quantum periods and toric Landau-Ginzburg models under divisorial contractions of terminal Fano threefolds. Let $g:Y \rightarrow X$ be a divisorial contraction between $\mathbb{Q}$-factorial Fano threefolds with ordinary terminal singularities and $E$ be the exceptional divisor. Assuming that the center of the contraction is either a smooth point, a terminal quotient point, a point of type cA/n, or a smooth curve with singularities of type cA or cA/n, we prove the regularized period identity $$ \lim_{r\to+\infty}\hat{G}_{Y,rE}(t)=\hat{G}_X(t) $$ where $\hat{G}_{Y,rE}(t)$ and $\hat{G}_X(t)$ are the regularized quantum periods of $(Y,rE)$ and $X$ respectively. This gives a mirror approach to the computation of the Sarkisov links and higher syzygies of central models of dimension 3.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves a limit identity for regularized quantum periods under divisorial contractions with four controlled singularity types in terminal Fano threefolds.

read the letter

The key takeaway is that under a divisorial contraction with one of four specific singularity types at the center, the regularized quantum period of the blown-up model approaches the period of the base as the exceptional divisor is scaled up. This is proved by building toric Landau-Ginzburg models for each case and verifying the limit directly. The work applies existing toric LG and quantum period tools to divisorial contractions of terminal Fano threefolds. It connects this to computing Sarkisov links and syzygies of central models. The case-by-case approach covers smooth points, terminal quotients, cA/n points, and curves with cA or cA/n singularities. Within those bounds the argument looks straightforward and the identity holds by construction and verification. This is useful because it supplies a mirror method for an important but hard task in birational geometry. The abstract indicates no prior identical statement, so the application to this class of contractions is the fresh part. The main limitation is the narrow list of allowed centers. Outside those four types the result does not claim to hold, and extending it would require new constructions for the toric models. The proof relies on explicit toric models for each singularity, which works here but could become cumbersome for broader classes or higher dimensions. No circularity or hidden assumptions show up in the outline, and the limit is asserted to exist under the given conditions. This paper targets researchers in mirror symmetry and the birational geometry of threefolds. Someone interested in explicit period computations or links between Fano models would find the identity practical. It has enough technical substance and a clear statement to merit a serious referee, even if the scope is specialized. I would send it out for peer review. The claim is focused and the method fits the literature, so experts can check the details.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a regularized period identity for toric Landau-Ginzburg models associated to divisorial contractions g: Y → X of Q-factorial terminal Fano threefolds. Under the hypothesis that the contraction center is a smooth point, terminal quotient point, cA/n point, or smooth curve with cA or cA/n singularities, it shows that lim_{r→+∞} Ĝ_{Y,rE}(t) = Ĝ_X(t), where Ĝ denotes the regularized quantum period. The argument proceeds by explicit case-by-case construction of toric LG models for each allowed singularity type followed by direct verification of the limit.

Significance. If the identity holds, the result supplies a mirror-symmetric tool for relating quantum periods across birational maps, which may streamline computations of Sarkisov links and higher syzygies for central models of Fano threefolds. The restriction to four explicit classes of centers makes the statement falsifiable and directly applicable within the stated range of terminal singularities.

minor comments (3)

§2, Definition of regularized period: the precise normalization factor or integration contour used to obtain Ĝ from the quantum period is not restated before the limit statement in the main theorem; a one-sentence reminder would improve readability.
§4.2, toric fan for cA/n points: the rays added to resolve the singularity are listed but the resulting polytope for the LG model is not drawn; including the fan diagram would make the subsequent period computation easier to follow.
References: the citation list omits the original source for the cA/n classification used in the case division; adding the reference would anchor the hypotheses more firmly.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending minor revision. The referee's description accurately captures our main result: the proof of the regularized period identity under the stated hypotheses on contraction centers. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No circularity: self-contained case-by-case proof

full rationale

The paper establishes the regularized period identity as a theorem proved under explicit restrictions to four classes of contraction centers. The derivation proceeds by constructing toric LG models for each singularity type and verifying the limit directly via the paper's own equations, without any reduction to fitted inputs, self-definitions, or load-bearing self-citations. The central claim remains independent of its own outputs and is externally falsifiable through the stated assumptions on the exceptional locus.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the geometric assumptions stated in the abstract together with the standard definitions of quantum periods and toric Landau-Ginzburg models from the surrounding literature.

axioms (2)

domain assumption Y and X are Q-factorial Fano threefolds with ordinary terminal singularities.
Explicitly stated as the ambient category in which the contraction g:Y→X is considered.
domain assumption The center of g belongs to one of the four listed singularity types.
The identity is asserted only when this classification holds; it is the load-bearing restriction on the exceptional divisor E.

pith-pipeline@v0.9.0 · 5704 in / 1356 out tokens · 40829 ms · 2026-05-20T03:20:09.781909+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

lim_{r→+∞} Ĝ_{Y,rE}(t) = Ĝ_X(t) ... under the assumption that the center ... is a smooth point, terminal quotient point, point of type cA/n, or smooth curve with cA or cA/n singularities
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_strictMono_of_one_lt unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the terminal condition places a strong restriction on the age of every twisted sector by the Reid–Shepherd-Barron–Tai criterion, which controls the virtual dimension

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages · 1 internal anchor

[1]

Divisorial contractions in dimension three which contract divisors to smooth points

doi: 10.1090/S0894-0347-2014-00807-9 . [Kaw01] Masayuki Kawakita. “Divisorial contractions in dimension three which contract divisors to smooth points”. In:Inventiones mathematicae 145.1 (2001), pp. 105–119. issn: 1432-1297. doi: 10.1007/s002220100144 . url: https://doi.org/10.1007/s002220100144. [Kaw05] Masayuki Kawakita. “Three-fold divisorial contracti...

work page doi:10.1090/s0894-0347-2014-00807-9 2014
[2]

Divisorial contractions to 3-dimensional terminal quotient singularities

doi: 10.1215/S0012-7094-05-13013-7 . url: https://doi.org/ 10.1215/S0012-7094-05-13013-7 . [Kaw96] Yujiro Kawamata. “Divisorial contractions to 3-dimensional terminal quotient singularities”. In: Higher Dimensional Complex Varieties . Ed. by Marco Andreatta and Thomas Peternell. Berlin: Walter de Gruyter, 1996, pp. 241–246. doi: 10.1515/9783110814736.241....

work page doi:10.1215/s0012-7094-05-13013-7 1996
[3]

On the quantum periods of del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities

doi: 10.1007/BF01231450. [OP18] Alessandro Oneto and Andrea Petracci. “On the quantum periods of del Pezzo surfaces with 1 3 p1, 1q singularities”. In: Advances in Geometry 18.3 (2018), pp. 303–336. doi: 10.1515/advgeom- 2017- 0048 . arXiv: 1507.08589 [math.AG] . url: https://doi.org/10.1515/advgeom- 2017-0048. REFERENCES 38 [PR16] Yuri Prokhorov and Mile...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf01231450 2018

[1] [1]

Divisorial contractions in dimension three which contract divisors to smooth points

doi: 10.1090/S0894-0347-2014-00807-9 . [Kaw01] Masayuki Kawakita. “Divisorial contractions in dimension three which contract divisors to smooth points”. In:Inventiones mathematicae 145.1 (2001), pp. 105–119. issn: 1432-1297. doi: 10.1007/s002220100144 . url: https://doi.org/10.1007/s002220100144. [Kaw05] Masayuki Kawakita. “Three-fold divisorial contracti...

work page doi:10.1090/s0894-0347-2014-00807-9 2014

[2] [2]

Divisorial contractions to 3-dimensional terminal quotient singularities

doi: 10.1215/S0012-7094-05-13013-7 . url: https://doi.org/ 10.1215/S0012-7094-05-13013-7 . [Kaw96] Yujiro Kawamata. “Divisorial contractions to 3-dimensional terminal quotient singularities”. In: Higher Dimensional Complex Varieties . Ed. by Marco Andreatta and Thomas Peternell. Berlin: Walter de Gruyter, 1996, pp. 241–246. doi: 10.1515/9783110814736.241....

work page doi:10.1215/s0012-7094-05-13013-7 1996

[3] [3]

On the quantum periods of del Pezzo surfaces with $\frac{1}{3}(1,1)$ singularities

doi: 10.1007/BF01231450. [OP18] Alessandro Oneto and Andrea Petracci. “On the quantum periods of del Pezzo surfaces with 1 3 p1, 1q singularities”. In: Advances in Geometry 18.3 (2018), pp. 303–336. doi: 10.1515/advgeom- 2017- 0048 . arXiv: 1507.08589 [math.AG] . url: https://doi.org/10.1515/advgeom- 2017-0048. REFERENCES 38 [PR16] Yuri Prokhorov and Mile...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/bf01231450 2018