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Smoothing Gorenstein toric Fano 3-folds

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Pith's one-line read Admissible Minkowski decomposition data on a 3-dimensional reflexive polytope determine a smoothing of the associated Gorenstein toric Fano 3-fold, and the number of ordinary double points in that smoothing is read off directly from the…

desk verdict A genuinely global smoothing construction for Gorenstein toric Fano 3-folds under a stated, restrictive combinatorial hypothesis; the main theorems hold up and it deserves a serious referee. read the letter →

arxiv 2412.06500 v1 pith:KUHS2MZH submitted 2024-12-09 math.AG

Alessio Corti , Paul Hacking , Andrea Petracci This is my paper · ORCID
classification math.AG MSC 14M2514J4514B0714J33
keywords toricFano3-foldGorensteinsingularitiessmoothingreflexivepolytopeMinkowskidecompositiontropicalarrangementBettinumbersmirrorsymmetry
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

This paper proves a sufficient combinatorial condition, called admissible Minkowski decomposition data (amd), for a Gorenstein toric Fano 3-fold to admit a smoothing, that is, a flat deformation to a less singular Fano 3-fold. The condition is purely polytopal: each facet of the reflexive polytope $P$ is decomposed into simple triangular pieces called A-triangles, and compatible tropical curve arrangements are chosen. From these data the paper constructs a toric partial resolution $Y$ of $X_P$ whose only singularities are quasi-ordinary double points, then shows $Y$ is smoothable and that a general smoothing blows down to a Fano 3-fold $X_t$ with ordinary double points. The paper also gives an effective recipe for the number of these double points—and hence the Betti numbers of the smoothing—directly from the combinatorial data. A consequence is that the same data that produce a Minkowski polynomial on the mirror side also produce the smooth Fano 3-fold on the A-side, explaining experimentally observed period equalities.

What carries the argument

The central objects are admissible Minkowski decomposition data (amd): for every facet $F$ of $P$, a Minkowski decomposition $F = \sum_j F_j$ into A-triangles (lattice triangles equivalent to $[(0,0),(0,1),(n+1,1)]$ with $n \ge -1$), together with a dual tropical arrangement subordinated to it; along each dull edge (colength one) the partitions of the edge's unit segments induced by the two incident facets must meet in sets of size at most one. From the tropical arrangements one builds an induced toric partial resolution $\pi:Y \to X$ with qODP singularities. The proof runs through the deformation theory of the qODP stack pair $(Y,E)$: its tangent space $T^1_{Y,E}$ is computed explicitly as sections on the partial normalization $\Delta'$ of the singular locus of $E$, and the matching condition guarantees a general Kodaira–Spencer class separates the unit segments. A global homogeneous (torus-equivariant) deformation, assembled from local Altmann deformations, is then shown to blow down to a deformation of $X$, and local-invariance results under rearrangement of tropical curves show the count of ODPs is combinatorial.

What would settle it

Run the construction on a specific 3-dimensional reflexive polytope with amd and inspect the fan of the induced partial resolution $Y$: every maximal cone must be either a basic simplex or the cone over a unit parallelogram; if any cone is otherwise, Lemma 1.14 fails and the smoothing argument collapses. Equivalently, compute a general Kodaira–Spencer section $s$ on the partial normalization $\Delta'$; if for some edge $e$ and indices $i<j$ the restriction difference $s_i - s_j$ has a multiple zero on the rational curve $\Delta^e_j$, then the predicted isolated ODP at that point is replaced by a worse singularity, contradicting Theorem 7.2.

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Extended reading notes

Core claim

Theorem 1.17 and Theorem 1.20 state: if a 3-dimensional reflexive polytope $P$ is endowed with amd, then the Gorenstein toric Fano 3-fold $X_P$ whose fan is the spanning fan of $P$ has a smoothing $X_t$ which is a Fano 3-fold. More precisely, the induced partial resolution $\pi: Y \to X$ has at worst quasi-ordinary double points (qODPs, the toric analogue of nodes), the pair $(Y,E)$ with its toric boundary is an unobstructed qODP stack pair, and a general smoothing $Y_t$ blows down to $X_t$, contracting finitely many disjoint nonsingular rational curves with normal bundle $O(-1) \oplus O(-1)$. The singularities of $X_t$ are exactly ordinary double points, and their number is $n = \sum_{\ell_e \ge 2} n_e$, where for each edge $e$ with incident facets $F,G$ and induced partitions $(L_i^F)$, $(L_j^G)$ of the $\ell_e$ unit segments, $n_e = k_e \binom{\ell_e}{2} - \sum_i \binom{|L_i^F|}{2} - \sum_j \binom{|L_j^G|}{2}$, with $k_e$ the colength. This gives the Betti numbers of the smoothing in terms of the combinatorics of $P$.

Load-bearing premise

The load-bearing premise is the matching condition on dull edges: for every edge of colength one, the two partitions of its unit segments induced by the two adjacent facets must intersect pairwise in at most one segment; if this fails, the general Kodaira–Spencer class may not separate the unit segments, and the resulting singularities need not be isolated ordinary double points.

Editorial extensions

If this is right

  • Every 3-dimensional reflexive polytope with amd gives a smoothing of its Gorenstein toric Fano 3-fold to a Fano 3-fold with ordinary double points.
  • The number of ODPs is $n = \sum_{\ell_e \ge 2} n_e$, with $n_e$ computed from edge lengths, colengths, and the sizes of the induced partitions; this determines the Betti numbers of the smoothing.
  • The smoothing $X_t$ contracts finitely many disjoint rational curves with normal bundle $O(-1) \oplus O(-1)$ above the ODPs, and $X_t$ is a deformation of the original $X_P$.
  • The same amd data that define a Minkowski polynomial (the B-side of mirror symmetry) now construct the smoothing (the A-side), and observed period equalities follow, though a conceptual explanation of the equality is still missing.
  • The companion computer study can place millions of amd-smoothings in the Mori–Mukai classification of Fano 3-folds.

Reading between the lines

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

  • The paper leaves open whether the amd condition is also necessary: because Remark 1.19 gives a smoothable polytope with no amd, the true boundary of the smoothing locus is likely a weaker condition that still separates unit segments along dull edges.
  • The explicit node count gives a fast invariant to distinguish smoothing components: two amd on the same polytope that yield different $n$ must produce non-isomorphic smoothings, while equal $n$ is consistent with deformation equivalence (as the paper notes for Betti numbers).
  • The qODP stack-pair technology is dimension-agnostic in its deformation-theoretic core, so the same strategy could yield sufficient smoothing conditions for Gorenstein toric Fano 4-folds, where the classification is unfinished.
  • A testable extension: compute $n = \sum n_e$ over the 4,319 reflexive 3-polytopes and compare the resulting Betti numbers against the known Hodge numbers of Fano 3-folds; any mismatch would pinpoint a polytope where the amd smoothing is not deformation-equivalent to the expected family.
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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

0 major / 4 minor

Summary. The paper introduces admissible Minkowski decomposition data (amd) for 3-dimensional reflexive polytopes, combining facet-wise admissible Minkowski decompositions into A-triangles with dual tropical arrangements satisfying a matching condition along dull edges. For a Gorenstein toric Fano 3-fold X whose fan is the spanning fan of P, the amd determine a toric partial resolution π: Y → X with only quasi-ordinary double points; the authors prove that the qODP stack pair (Y,E) is unobstructed and smoothable, and that the induced deformation blows down to a deformation X_t of X whose general member is a Fano 3-fold with ordinary double points. The second main result gives an explicit combinatorial formula for the number n of ODPs and hence for the Betti numbers of the smoothing, computed from the induced partitions of the unit segments of each edge of P. The central theorems (1.17 and 1.20) are conditional on the existence of amd, a restriction the authors explicitly flag as non-optimal in Remark 1.19.

Significance. If correct, the paper provides a systematic, purely combinatorial construction of smoothings for a large class of Gorenstein toric Fano 3-folds and an exact recipe for their Betti numbers. The Betti number formula is parameter-free: it is derived from the combinatorics of the reflexive polytope rather than fitted to known examples, and the paper gives falsifiable predictions that are tested in the companion paper [11]. The exposition is honest about the scope of the method: Remark 1.19 states that the result is not optimal and that some smoothable polytopes admit no amd, and the footnote on page 2 acknowledges that the dependence of the smoothing on the choice of dual tropical arrangement is not fully understood beyond Betti numbers. The proof is organised through substantial appendices (deformation theory of pairs, miniversal families, Rim's equivariant G-structures) that make the arguments largely self-contained.

minor comments (4)
  1. [Section 6, proof of Theorem 6.1] The assertion that any two generic dual tropical arrangements can be joined by a sequence of moves of type I, their inverses, and type II–IV is stated without proof or reference. Since this connectivity is used to reduce to Lemma 6.2, please add a brief justification (for example via the connectivity of the graph of regular triangulations of the Cayley polytope, cf. Theorem A.6 and [51]) or a precise citation.
  2. [Lemma 1.14] The proof that the cones of the fan of Y are either basic simplices or cones over unit parallelograms is quite terse; a short explanation in terms of the fine mixed subdivision and the dual tropical arrangement (Theorem A.6) would make the lemma easier to verify.
  3. [Theorem 1.20 and Remark 7.3(2)] The count leading to the formula for n_e is described as 'a simple exercise'; since this formula is the main computational output, a few lines deriving it from the degree-k_e zeros of s_i - s_j (with the subtractions of the contributions at the vertices coming from the parts of the induced partitions) would improve the paper's utility.
  4. [Page 2 footnote] The footnote on page 2 flags, correctly, that the authors do not know whether different dual tropical arrangements satisfying the matching condition lead to deformation equivalent smoothings, while Betti numbers coincide. This limitation is transparent, but it is worth stating explicitly in the introduction's summary of Theorem 1.17 that the smoothing is determined by the amd up to the specified ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the smoothing and Betti-number theorems are derived from amd data, not fitted or self-referential.

full rationale

The derivation chain is self-contained under the stated amd hypotheses. The smoothing is obtained from a global Altmann deformation (Proposition 2.12), whose construction is quoted from [44] but is an external published theorem with assumptions that do not include the target result; it is used as a tool, not as a substitute for the proof of Theorem 1.17. The matching condition (Definition 1.10(c)) is a genuine hypothesis: Corollary 3.6 uses it to ensure a general Kodaira-Spencer class separates unit segments, and this feeds into Lemma 5.2 and Theorem 7.2; nowhere is the conclusion read back into the hypothesis. The ODP count n_e in Theorem 1.20 is a computed output from the induced partitions, not a fitted parameter or a renamed input. Remark 1.19 explicitly labels the theorem non-optimal, and the footnote in Section 1.1 concedes an unproved deformation-equivalence statement; both are limitations, not circular steps. No uniqueness theorem from the authors is invoked to force a choice, and no empirical pattern is renamed as a prediction. Therefore no circular step was found.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

Pure mathematics paper; no numerical constants are fitted to data. The combinatorial choices in an amd are discrete data, not free parameters, and the Betti number formula is derived from the definitions. The matching condition is a new ad hoc hypothesis, explicitly admitted to be non-optimal.

assumptions (5)
  • standard math The correspondence between 3-dimensional reflexive polytopes and Gorenstein toric Fano 3-folds via the spanning fan.
    Used throughout: X = X_P is the spanning fan of P; this is standard toric geometry (Section 1.1).
  • standard math Altmann's explicit description of the infinitesimal deformation space T^1 of 3-dimensional Gorenstein toric cones ([4, §4.3]).
    Used in Lemma 2.17 to compute T^1_{X,D}; cited from Altmann's paper.
  • standard math Rim's theorem on T-equivariant structures on versal deformations.
    Appendix D, used in the proof of Proposition 2.14 to obtain formal lifts.
  • standard math Existence of log terminal models for projective morphisms between complex analytic spaces ([20, Theorem 1.2]).
    Used in proof of Lemma 6.2 (Step 1) to construct the birational map between Y and Y' as a log terminal model over X.
  • ad hoc to paper The matching condition at dull edges, imposed as part of the definition of amd.
    Definition 1.10(c). It is a sufficient condition for the Kodaira-Spencer class to have distinct values on unit segments (Corollary 3.6), hence for the smoothing to have isolated ODPs; the paper acknowledges it is not necessary (Remark 1.19(2)) and cites counterexamples when it fails.
invented entities (1)
  • Admissible Minkowski decomposition data (amd) independent evidence
    purpose: Combinatorial structure on a 3D reflexive polytope P used to construct a smoothing of the Gorenstein toric Fano 3-fold X_P and to compute Betti numbers of the smoothing.
    New mathematical definition, not a physical entity. It is falsifiable in the mathematical sense: the resulting smoothing's Betti numbers and ODP count can be compared against the known classification of Fano 3-folds; the companion paper [11] carries out such checks for millions of amd.

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Pith. "Pith review of Smoothing Gorenstein toric Fano 3-folds." pith.science (2026). https://pith.science/paper/KUHS2MZH

@misc{pith2026241206500,
  author       = {Pith},
  title        = {Pith review of: Smoothing Gorenstein toric Fano 3-folds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KUHS2MZH}},
  note         = {Machine review of arXiv:2412.06500}
}
read the original abstract

We introduce admissible Minkowski decomposition data (amd) for a 3-dimensional reflexive polytope P. This notion is defined purely in terms of the combinatorics of P. Denoting by X the Gorenstein toric Fano 3-fold whose fan is the spanning fan (a.k.a. face fan) of P, our first result states that amd for P determine a smoothing of X. Our second result amounts to an effective recipe for computing the Betti numbers of the smoothing.

Figures

Figures reproduced from arXiv: 2412.06500 by the authors.

Figure 1
Figure 1. An A−1-triangle, an A0-triangle, an A1-triangle and an A2-triangle. Definition 1.3. The Minkowski sum of lattice polytopes F1, . . . , Fr in a lattice L is the lattice poly￾tope† F1 + · · · + Fr = {v1 + · · · + vr | v1 ∈ F1, . . . , vr ∈ Fr} A Minkowski decomposition of a lattice polytope F is a tuple of lattice polytopes whose Minkowski sum is F. These lattice polytopes are called Minkowski summands of F. Definitio… view at source ↗
Figure 2
Figure 2. The two admissible Minkowski decompositions of the hexagon in Example 1.5. Example 1.6. Some polygons have no admissible Minkowski decompositions, e.g. the triangle [(−1, −1),(2, −1),(−1, 1)] and the triangle [(−1, −1),(1, 0),(0, 1)] in Z 2 , see [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. These polygons have no admissible Minkowski decomposition. Definition 1.7. Fix a plane polygon F and admissible Minkowski decomposition m = (F = PFj ). To each Minkowski summand Fj is attached a dual tropical curve Γj, see Theorem A.6 and its proof. A dual tropical arrangement subordinated to m is a generic plane arrangement of the Γj . A dual tropical arrangement induces, for every edge e ⊂ F, a partition of the se… view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Dual tropical arrangements and induced subdivisions for the admissible Minkowski decomposition of the hexagon into two A0-triangles. Definition 1.9. Let P be a 3-dimensional reflexive polytope, and e ≤ P an edge of P. The length of e, denoted by ℓe, is the integral len…
Figure 5
Figure 5. Figure 5: Dual tropical arrangements and induced subdivisions for the admissible Minkowski decomposition of the hexagon into three A−1-triangles [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: More dual tropical arrangements and induced subdivisions. denote by L F i the part corresponding to Fi . The second is induced by the Minkowski decomposition G = PGj , and we denote by L G j the part corresponding to Gj . The matching condition at e is the following st…
Figure 7
Figure 7. Figure 7: Exceptional surfaces over transverse A3. x4 x3 x1 x2 ` ν −−−−−→ x1 x2 x4 x3 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: The partial normalization ν : ∆′ → ∆. For all edges e ≤ P, we denote by ℓe the (lattice) length of e, by e ⋆ = [v, w] ≤ Q the dual edge, and by ke the colength of e, that is, the length of e ⋆ . We denote by Γe ⊂ X the closure of the 1-dimensional torus orbit correspon…
Figure 9
Figure 9. Figure 9: Move of Type I. µ + uD1(u) λ + uD0(u) λ λ + xA(x) µ + zC(z) λ + yB(y) 99K λ + uD0(u) µ + uD1(u) λ λ + xA(x) µ + zC(z) λ + yB(y) [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]
Figure 10
Figure 10. Figure 10: Move of Type II. λ1 + xA(x) λ2 + zC(z) λ2 λ3 λ1 λ2 + uD(u) λ3 + vE(v) λ1 + yB(y) λ3 + wF(w) 99K λ1 + xA(x) λ2 + zC(z) λ2 + uD(u) λ3 + vE(v) λ1 + yB(y) λ3 λ2 λ1 λ3 + wF(w) [PITH_FULL_IMAGE:figures/full_fig_p030_10.png]
Figure 11
Figure 11. Figure 11: Move of type III deformations of (Y, E) in terms of local data in § C.2, we may replace X by the toric affine chart at a 0-stratum of Z, so that we are in the situation of Lemma 6.2. The statement then follows from Lemma 6.2 [PITH_FULL_IMAGE:figures/full_fig_p030_11.png]
Figure 12
Figure 12. Figure 12: Move of type IV Note that Y → X and Y ′ → X have relative Picard rank 1 and we are assuming that Π is small. Let A be a Π-ample divisor on Y. Let H be a hyperplane section of X containing π∗A. Let ∆ be the effective Cartier divisor ǫ(π ∗H − A) for 0 < ǫ ≪ 1, ǫ ∈ Q. Th…
Figure 13
Figure 13. Figure 13: Exceptional curve and tangent vector ξ ∈ T0M = H0 [PITH_FULL_IMAGE:figures/full_fig_p034_13.png]
Figure 14
Figure 14. Figure 14: Kodaira–Spencer map of the family of Equations C.1 and C.2. Example C.10. In this example we describe a miniversal deformation of the toric pair (Y, E) where Y is the product of the minimal resolution of the surface An-singularity with A 1 . In more detail, consider t…
Figure 15
Figure 15. Figure 15: Kodaira–Spencer map of the family of Equations C.3 and C.4. (iv) The action of the symmetric group Sn+1 that permutes the factors of (A∞) n+1 lifts uniquely to: • a biregular action on (X, D) such that f : (X, D) → M is equivariant, and • a birational action on the pa…
Figure 16
Figure 16. Figure 16: Tropical arrangement for the pair (Y, E) of Example C.12. (i) The rational map (Y, E) 99K (Y, E) is a morphism that identifies (Y, E) with the fibre g ⋆ (0); (ii) With the identification in Part (i), g : (Y, E) → 0 ∈ M is a miniversal family for the pair (Y, E); (iii)…
Figure 17
Figure 17. Figure 17: Kodaira–Spencer map of the family of Equations C.5 and C.6. (iv) Let Π′ : Y ′ → X be the blow-up of the divisor Z ′ =  y = z − D0(u) = 0 ⊂ X and E ′ = Π′ −1 (D). Then g ′ = f ◦ Π′ : (Y ′ , E ′ ) → M is a miniversal family for the pair (Y ′ , E′ ) given by the tropic…
Figure 18
Figure 18. Figure 18: Kodaira–Spencer map of the family of Claim C.13. where the important thing is that we can solve for y, that is, Y is nonsingular in this chart. The divisor E is given by: u [PITH_FULL_IMAGE:figures/full_fig_p052_18.png]

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Works this paper leans on

63 extracted references · 46 canonical work pages · cited by 1 Pith paper

  1. [11]

    On t he topology of Fano smoothings

    Tom Coates, Alessio Corti, and Genival da Silva, Jr. On t he topology of Fano smoothings. In Interactions with lattice polytopes , volume 386 of Springer Proc. Math. Stat. , pages 135–156. Springer, Cham, [2022] ©2022. doi:10.1007/978-3-030-98327-7_6

    work page doi:10.1007/978-3-030-98327-7_6 2022

  2. [1]

    Kasprzyk

    Mohammad Akhtar, Tom Coates, Sergey Galkin, and Alexand er M. Kasprzyk. Minkowski polynomials and muta- tions. SIGMA Symmetry Integrability Geom. Methods Appl. , 8:Paper 094, 17, 2012. doi:10.3842/SIGMA.2012.094

    work page doi:10.3842/sigma.2012.094 2012

  3. [2]

    Minkowski sums and homogeneous deformat ions of toric varieties

    Klaus Altmann. Minkowski sums and homogeneous deformat ions of toric varieties. Tohoku Math. J. (2) , 47(2):151– 184, 1995. doi:10.2748/tmj/1178225590. SMOOTHING GORENSTEIN TORIC F ANO 3-FOLDS 53

    arXiv 1995

  4. [3]

    Infinitesimal deformations and obstruct ions for toric singularities

    Klaus Altmann. Infinitesimal deformations and obstruct ions for toric singularities. J. Pure Appl. Algebra , 119(3):211–235, 1997. doi:10.1016/S0022-4049(96)00029-1

    work page doi:10.1016/s0022-4049(96)00029-1 1997

  5. [4]

    One parameter families containing three -dimensional toric Gorenstein singularities

    Klaus Altmann. One parameter families containing three -dimensional toric Gorenstein singularities. In Explicit birational geometry of 3-folds , volume 281 of London Math. Soc. Lecture Note Ser. , pages 21–50. Cambridge Univ. Press, Cambridge, 2000

    work page 2000

  6. [5]

    Deforming Stanley–Reisner schemes

    Klaus Altmann and Jan Arthur Christophersen. Deforming Stanley–Reisner schemes. Math. Ann., 348(3):513–537,

    work page

  7. [6]

    Polyhedral divisors and algebraic torus actions

    Klaus Altmann and J¨ urgen Hausen. Polyhedral divisors and algebraic torus actions. Math. Ann. , 334(3):557–607,

    work page

  8. [7]

    The geometry of T - varieties

    Klaus Altmann, Nathan Owen Ilten, Lars Petersen, Hendri k S¨ uß, and Robert Vollmert. The geometry of T - varieties. In Contributions to algebraic geometry , EMS Ser. Congr. Rep., pages 17–69. Eur. Math. Soc., Z¨ urich ,

    work page

Show all 63 references
  1. [8]

    On the solutions of analytic equations

    Michael Artin. On the solutions of analytic equations. Invent. Math. , 5:277–291, 1968. doi:10.1007/BF01389777

    1968 doi

  2. [9]

    Lectures on deformations of singularities , volume 54

    Michael Artin. Lectures on deformations of singularities , volume 54. Tata Institute of Fundamental Research Bom- bay, 1976

    1976

  3. [10]

    Hacon, a nd James McKernan

    Caucher Birkar, Paolo Cascini, Christopher D. Hacon, a nd James McKernan. Existence of minimal models for varieties of log general type. J. Amer. Math. Soc. , 23(2):405–468, 2010. doi:10.1090/S0894-0347-09-00649-3

    2010 doi

  4. [12]

    Quantum periods for 3-dimensional Fano manifolds

    Tom Coates, Alessio Corti, Sergey Galkin, and Alexande r Kasprzyk. Quantum periods for 3-dimensional Fano manifolds. Geom. Topol., 20(1):103–256, 2016. doi:10.2140/gt.2016.20.103

    2016 doi

  5. [13]

    Kasprzyk, Giuseppe Pitton, and Ketil Tveiten

    Tom Coates, Alexander M. Kasprzyk, Giuseppe Pitton, and Ketil Tveiten. Maximally mutable Laurent polynomials. Proc. A., 477(2254):Paper No. 20210584, 21, 2021

    2021

  6. [14]

    Mirro r symmetry and smoothing Gorenstein toric affine 3-folds

    Alessio Corti, Matej Filip, and Andrea Petracci. Mirro r symmetry and smoothing Gorenstein toric affine 3-folds. In Facets of algebraic geometry. Vol. I , volume 472 of London Math. Soc. Lecture Note Ser. , pages 132–163. Cambridge Univ. Press, Cambridge, 2022

    2022

  7. [15]

    How to make log structur es

    Alessio Corti and Helge Ruddat. How to make log structur es. arXiv e-prints , December 2023. arXiv:2312.13867, doi:10.48550/arXiv.2312.13867

    2023 doi

  8. [16]

    De Loera, J¨ org Rambau, and Francisco Santos

    Jes´ us A. De Loera, J¨ org Rambau, and Francisco Santos. Triangulations, volume 25 of Algorithms and Computation in Mathematics . Springer-Verlag, Berlin, 2010. Structures for algorithm s and applications. doi:10.1007/978-3-642-12971-1

    2010 doi

  9. [17]

    Le probl` eme des modules locaux pour lesespaces C-analytiques compacts

    Adrien Douady. Le probl` eme des modules locaux pour lesespaces C-analytiques compacts. Ann. Sci. ´Ecole Norm. Sup. (4) , 7:569–602, 1974. URL: http://www.numdam.org/item?id=ASENS_1974_4_7_4_569_0

    1974

  10. [18]

    Commutative algebra, volume 150 of Graduate Texts in Mathematics

    David Eisenbud. Commutative algebra, volume 150 of Graduate Texts in Mathematics . Springer-Verlag, New York,

  11. [19]

    Global smoothings of varieties with n ormal crossings

    Robert Friedman. Global smoothings of varieties with n ormal crossings. Ann. of Math. (2) , 118(1):75–114, 1983. doi:10.2307/2006955

    1983 doi

  12. [20]

    Minimal model program for projective mor phisms between complex analytic spaces

    Osamu Fujino. Minimal model program for projective mor phisms between complex analytic spaces. arXiv e-prints , page arXiv:2201.11315, January 2022. arXiv:2201.11315, doi:10.48550/arXiv.2201.11315

    2022 doi arXiv

  13. [21]

    Introduction to toric varieties , volume 131 of Annals of Mathematics Studies

    William Fulton. Introduction to toric varieties , volume 131 of Annals of Mathematics Studies . Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures i n Geometry. doi:10.1515/9781400882526

    1993 doi

  14. [22]

    A study in derived algebraic geometry

    Dennis Gaitsgory and Nick Rozenblyum. A study in derived algebraic geometry. Vol. II. Deformation s, Lie theory and formal geometry , volume 221 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2017. doi:10.1090/surv/221.2

    2017 doi

  15. [23]

    Singular symplec tic spaces and holomorphic membranes

    Sergey Galkin and Grigory Mikhalkin. Singular symplec tic spaces and holomorphic membranes. Eur. J. Math. , 8(3):932–951, 2022. doi:10.1007/s40879-022-00568-y

    2022 doi

  16. [24]

    Der Satz von Kuranishi f¨ ur kompakte komp lexe R¨ aume

    Hans Grauert. Der Satz von Kuranishi f¨ ur kompakte komp lexe R¨ aume. Invent. Math. , 25:107–142, 1974. doi:10.1007/BF01390171

    1974 doi

  17. [25]

    Mirror symmetry via logar ithmic degeneration data

    Mark Gross and Bernd Siebert. Mirror symmetry via logar ithmic degeneration data. I. J. Differential Geom. , 72(2):169–338, 2006. URL: http://projecteuclid.org/euclid.jdg/1143593211

    2006

  18. [26]

    ´El´ ements de g´ eom´ etrie alg´ ebrique

    Alexander Grothendieck. ´El´ ements de g´ eom´ etrie alg´ ebrique. IV. ´Etude locale des sch´ emas et des morphismes de sch´ emas IV. Inst. Hautes ´Etudes Sci. Publ. Math. , 32:5–361, 1967. URL: http://www.numdam.org/item/PMIHES_1967__32__5_0

    1967

  19. [27]

    Local cohomology.Springer-Verlag, Berlin-New York,, 1967

    Robin Hartshorne. Local cohomology.Springer-Verlag, Berlin-New York,, 1967. A seminar given by A. Grothendieck, Harvard University, Fall, 1961

    1967

  20. [28]

    The Cayley trick, lifting subdivisions and the Bohne–Dress theorem on zonotopal tilings

    Birkett Huber, J¨ org Rambau, and Francisco Santos. The Cayley trick, lifting subdivisions and the Bohne–Dress theorem on zonotopal tilings. J. Eur. Math. Soc. (JEMS) , 2(2):179–198, 2000. doi:10.1007/s100970050003

    doi

  21. [29]

    Complexe cotangent et d´ eformations

    Luc Illusie. Complexe cotangent et d´ eformations. I. Springer-Verlag, Berlin-New York,,, 1971

    1971

  22. [30]

    Mutations of Laurent polynomials an d flat families with toric fibers

    Nathan Owen Ilten. Mutations of Laurent polynomials an d flat families with toric fibers. SIGMA Symmetry Inte- grability Geom. Methods Appl. , 8:Paper 047, 7, 2012. doi:10.3842/SIGMA.2012.047. 54 ALESSIO CORTI, PAUL HACKING, AND ANDREA PETRACCI

    2012 doi

  23. [31]

    Deformations of rational T -varieties

    Nathan Owen Ilten and Robert Vollmert. Deformations of rational T -varieties. J. Algebraic Geom. , 21(3):531–562,

  24. [32]

    The Cayley trick for tropical hypersurfaces with a view toward Ricardian economics

    Michael Joswig. The Cayley trick for tropical hypersurfaces with a view toward Ricardian economics. InHomological and computational methods in commutative algebra , volume 20 of Springer INdAM Ser. , pages 107–128. Springer, Cham, 2017

    2017

  25. [33]

    Essentials of tropical combinatorics , volume 219 of Graduate Studies in Mathematics

    Michael Joswig. Essentials of tropical combinatorics , volume 219 of Graduate Studies in Mathematics . American Mathematical Society, Providence, RI, [2021] ©2021. doi:10.1090/gsm/219

    2021 doi

  26. [34]

    Vertex algebras an d the formal loop space

    Mikhail Kapranov and Eric Vasserot. Vertex algebras an d the formal loop space. Publ. Math. Inst. Hautes ´Etudes Sci., 100:209–269, 2004. doi:10.1007/s10240-004-0023-9

    2004 doi

  27. [35]

    Flips, flops, minimal models, etc

    J´ anos Koll´ ar. Flips, flops, minimal models, etc. InSurveys in differential geometry (Cambridge, MA, 1990) , pages 113–199. Lehigh Univ., Bethlehem, PA, 1991

    1990

  28. [36]

    On the locally complete families of complex analytic structures

    Masatake Kuranishi. On the locally complete families of complex analytic structures. Ann. of Math. (2) , 75:536–577,

  29. [37]

    Birkh¨ auser Verlag, Basel, 1991

    Stanis/suppress law /suppress Lojasiewicz.Introduction to complex analytic geometry . Birkh¨ auser Verlag, Basel, 1991. Translated from the Polish by Maciej Klimek. doi:10.1007/978-3-0348-7617-9

    1991 doi

  30. [38]

    Commutative ring theory , volume 8 of Cambridge Studies in Advanced Mathematics

    Hideyuki Matsumura. Commutative ring theory , volume 8 of Cambridge Studies in Advanced Mathematics . Cam- bridge University Press, Cambridge, second edition, 1989. Translated from the Japanese by M. Reid

    1989

  31. [39]

    Simultaneous minimal models of homogeneous toric deformations

    Daisuke Matsushita. Simultaneous minimal models of homogeneous toric deformations. Kodai Math. J., 25(1):54–60,

  32. [40]

    Mavlyutov

    Anvar R. Mavlyutov. Deformations of Toric Varieties via Minkowski Sum Decompositions of Polyhedral Complexes. arXiv e-prints , February 2009. arXiv:0902.0967, doi:10.48550/arXiv.0902.0967

    2009 doi arXiv

  33. [41]

    Smoothing Fano 3-folds

    Yoshinori Namikawa. Smoothing Fano 3-folds. J. Algebraic Geom. , 6(2):307–324, 1997

    1997

  34. [42]

    Algebraic spaces and stacks , volume 62 of American Mathematical Society Colloquium Publications

    Martin Olsson. Algebraic spaces and stacks , volume 62 of American Mathematical Society Colloquium Publications . American Mathematical Society, Providence, RI, 2016. doi:10.1090/coll/062

    2016 doi

  35. [43]

    Some examples of non-smoothable Gore nstein Fano toric threefolds

    Andrea Petracci. Some examples of non-smoothable Gore nstein Fano toric threefolds. Math. Z. , 295(1-2):751–760,

  36. [44]

    Homogeneous deformations of toric pa irs

    Andrea Petracci. Homogeneous deformations of toric pa irs. Manuscripta Math. , 166(1-2):37–72, 2021. doi:10.1007/s00229-020-01219-w

    2021 doi

  37. [45]

    Lagrangian torus fibration models of Fan o threefolds

    Thomas Prince. Lagrangian torus fibration models of Fan o threefolds. arXiv e-prints , January 2018. arXiv:1801.02997, doi:10.48550/arXiv.1801.02997

    2018 doi arXiv

  38. [46]

    Cracked polytopes and Fano toric comple te intersections

    Thomas Prince. Cracked polytopes and Fano toric comple te intersections. Manuscripta Math. , 163(1-2):165–183,

  39. [47]

    From cracked polytopes to Fano threefol ds

    Thomas Prince. From cracked polytopes to Fano threefol ds. Manuscripta Math. , 164(1-2):267–320, 2021. doi:10.1007/s00229-020-01180-8

    2021 doi

  40. [48]

    Canonical 3-folds

    Miles Reid. Canonical 3-folds. In Journ´ ees de G´ eometrie Alg´ ebrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979 , pages 273–310. Sijthoff & Noordhoff, Alphen aan den Rijn—Ger mantown, Md., 1980

    1979

  41. [49]

    Dock S. Rim. Equivariant G-structure on versal deformations. Trans. Amer. Math. Soc. , 257(1):217–226, 1980. doi:10.2307/1998132

    1980 doi

  42. [50]

    Teresa Sancho, Jose P

    M. Teresa Sancho, Jose P. Moreno, and Carlos Sancho. Aut omorphism group of a toric variety. arXiv e-prints , September 2018. arXiv:1809.09070, doi:10.48550/arXiv.1809.09070

    2018 doi arXiv

  43. [51]

    The Cayley trick and triangulations of products of simplices

    Francisco Santos. The Cayley trick and triangulations of products of simplices. In Integer points in polyhedra— geometry, number theory, algebra, optimization , volume 374 of Contemp. Math. , pages 151–177. Amer. Math. Soc., Providence, RI, 2005. doi:10.1090/conm/374/06904

    2005 doi

  44. [52]

    Deformations of theta divi sors and the rank 4 quadrics problem

    Roy Smith and Robert Varley. Deformations of theta divi sors and the rank 4 quadrics problem. Compositio Math., 76(3):367–398, 1990. URL: http://www.numdam.org/item?id=CM_1990__76_3_367_0

    1990

  45. [53]

    doi:10.1007/s00229-019-01149-2

    doi

  46. [54]

    On the Newton polytope of the resultan t

    Bernd Sturmfels. On the Newton polytope of the resultan t. J. Algebraic Combin. , 3(2):207–236, 1994. doi:10.1023/A:1022497624378

    1994 doi

  47. [55]

    Jonathan M. Wahl. Equisingular deformations of normal surface singularities. I. Ann. of Math. (2) , 104(2):325–356,

  48. [56]

    Deformations of pairs ( X, L) when X is singular

    Jie Wang. Deformations of pairs ( X, L) when X is singular. Proc. Amer. Math. Soc. , 140(9):2953–2966, 2012. doi:10.1090/S0002-9939-2011-11230-9 . Department of Mathematics, Imperial College London, 180 Qu een’s Gate, London, SW7 2AZ, UK Email address : a.corti@imperial.ac.uk ...

    2012 doi

  49. [60]

    Stacks Project

    The Stacks Project Authors. Stacks Project. https://stacks.math.columbia.edu, 2018

    2018

  50. [1995]

    doi:10.1007/978-1-4612-5350-1

    With a view toward algebraic geometry. doi:10.1007/978-1-4612-5350-1

    doi

  51. [2002]

    doi:10.2996/kmj/1106171075

  52. [2006]

    doi:10.1007/s00208-005-0705-8

    doi

  53. [2010]

    doi:10.1007/s00208-010-0490-x

    doi

  54. [2012]

    doi:10.1090/S1056-3911-2011-00585-7

    2011 doi

  55. [2020]

    doi:10.1007/s00209-019-02369-8

    doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.