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Mirrors to toric degenerations via intrinsic mirror symmetry

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Pith's one-line read This paper proves that the intrinsic mirror construction, after resolving a special toric degeneration of K3 surfaces and base-changing by the polarization, yields a family isomorphic to the classical toric degeneration mirror.

desk verdict Strong thesis proving the K3 case of a new Gross-Siebert mirror conjecture, but the proof's load-bearing reduction rests on an unproved extension of Johnston's birational invariance theorem. read the letter →

arxiv 2608.07381 v1 pith:3AMPYQ4T submitted 2026-08-07 math.AG

Evgeny Goncharov This is my paper · ORCID
classification math.AG MSC 14J3314J2814N3514M2514T05
keywords mirrorsymmetrytoricdegenerationsintrinsicK3surfacesscatteringdiagramslogCalabi-YaupuncturedGromov-Witteninvariantssmoothresolutions
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

Mirror symmetry has two competing constructions for Calabi-Yau families with toric special fibres: an older algorithmic one that reconstructs a mirror from a scattering diagram built out of the combinatorial data of the degeneration, and a newer intrinsic one defined through punctured log Gromov-Witten invariants. This paper shows they agree for K3 surfaces. For any special toric degeneration of K3 surfaces with a polarization, the author constructs an admissible resolution to a minimal log Calabi-Yau degeneration and proves that the intrinsic mirror of the resolution, base-changed along the map sending a curve class to its intersection with the pulled-back polarization, is isomorphic to the toric degeneration mirror. The comparison is made by relating the two scattering diagrams that encode the two constructions. The paper also extends the correspondence to the minimal relative locus of the extended intrinsic mirror, matching the universal toric degeneration mirror, and constructs log smooth resolutions for a natural family of toric degenerations of Calabi-Yau threefolds.

What carries the argument

The load-bearing object is the scattering diagram: a finite set of walls, each carrying a function, on the dual intersection complex of a degeneration, encoding the curve-counting corrections that define a mirror family. The paper compares the algorithmic scattering diagram $\bar{\mathfrak D}$ of a toric degeneration with the canonical scattering diagram $\mathfrak D$ of its admissible log-smooth resolution. The canonical diagram is defined from wall types and punctured log Gromov-Witten invariants; the comparison, after the base change $h$, shows the two diagrams are combinatorially equivalent, and by the uniqueness property of the algorithmic construction this forces the two mirror families to be isomorphic. A second key device is the admissible resolution itself: blowing up components of the central fibre to make the degeneration log smooth and minimal log Calabi-Yau while keeping control of the dual intersection complex and curve classes.

What would settle it

Compute the two mirror families, or their defining scattering diagrams, modulo $t^2$ for a specific special toric degeneration of K3 surfaces such as the quartic hypersurface in $\mathbb P^3$ degenerating to the coordinate tetrahedron, and check whether the basechanged intrinsic mirror is isomorphic to the toric degeneration mirror; a mismatch in any leading-order wall or equation would disprove Theorem A.

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Extended reading notes

Core claim

The central discovery is Theorem A: Conjecture 1.7 holds for special toric degenerations of K3 surfaces. Concretely, every such degeneration $\bar{\mathfrak X}\to \mathcal S$ with polarization $A$ admits an admissible resolution $\pi:\mathfrak X\to \bar{\mathfrak X}$ to a minimal log Calabi-Yau degeneration, and the basechange of the intrinsic mirror $\check{\mathfrak X}\to \operatorname{Spf} \widehat{k[P]}$ by the homomorphism $h:\beta\mapsto \pi_* A\cdot \beta$ is isomorphic to the toric degeneration mirror $\check{\bar{\mathfrak X}}\to \operatorname{Spec} k[t]$. The proof constructs the resolution, interprets the extended intrinsic mirror through scattering diagrams, and proves that the canonical scattering diagram of the intrinsic construction becomes combinatorially equivalent to the algorithmic scattering diagram of the toric degeneration mirror after the base change. In addition, Theorem B upgrades this to an isomorphism between the restriction of the intrinsic mirror to the minimal relative Gross-Siebert locus and a subfamily of the universal toric degeneration mirror, and Theorem C provides strongly admissible resolutions for a class of toric degenerations of Calabi-Yau threefolds obtained by the reconstruction algorithm.

Load-bearing premise

The argument leans on an externally supplied result, birational invariance of punctured log Gromov-Witten invariants, which is used to reduce Conjecture 1.7 to strongly admissible resolutions and to remove a technical global-generation assumption; if that invariance fails, the reduction and the scattering-diagram comparison do not go through.

Editorial extensions

If this is right

  • For K3 surfaces, the intrinsic mirror construction is no longer a separate object: it recovers the toric degeneration mirror exactly, so results and algorithms for one construction transfer to the other.
  • The toric degeneration mirror is shown to depend only on the intended data — central fibre, polarization, and chosen slab data — by being identified with the intrinsic mirror after base change.
  • The extended correspondence over the minimal relative Gross-Siebert locus means the intrinsic mirror contains the universal family of the toric degeneration mirror, varying freely in initial slab functions and gluing data.
  • In dimension three, a natural class of toric degenerations admits, after a bounded base change, strongly admissible resolutions, giving a path toward the same comparison for Calabi-Yau threefolds.
  • For elliptic curves, the two mirror constructions agree tautologically and the universal toric degeneration mirror is isomorphic to the intrinsic mirror, giving the base case of the general picture.

Reading between the lines

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

  • If the comparison extends to higher dimensions, the intrinsic mirror would supply a canonical choice of the initial slab functions and gluing data that the algorithmic toric degeneration mirror currently requires as input, removing a source of non-uniqueness.
  • The reliance on birational invariance suggests the intrinsic mirror should be independent of the chosen admissible resolution; if true, it would be a canonical mirror attached to the original degeneration rather than to a resolution.
  • A concrete testable extension would be to carry out the scattering-diagram comparison explicitly for the quartic K3 in $\mathbb P^3$ and verify the basechanged canonical diagram equals the algorithmic one wall by wall through all orders in $t$.
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Editorial analysis

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Referee Report

4 major / 4 minor

Summary. The paper develops a comparison between two mirror constructions in the Gross-Siebert program: the algorithmic toric degeneration mirror and the intrinsic mirror. After an explicit treatment of degenerations of elliptic curves, the main result (Theorem A / Theorem 4.73) asserts that Conjecture 1.7 holds for special toric degenerations of K3 surfaces with smooth generic fibre: any such degeneration admits an admissible resolution to a minimal log Calabi-Yau degeneration, and the basechange of the intrinsic mirror by the polarization map is isomorphic to the toric degeneration mirror. The paper also proves a correspondence between the restriction of the intrinsic mirror to the minimal relative Gross-Siebert locus and the universal toric degeneration mirror (Theorem B / Theorem 5.35), and constructs log smooth resolutions for a natural family of toric degenerations of Calabi-Yau threefolds (Theorem C / Theorem 6.16). The argument is organized around reducing to strongly admissible resolutions, removing the simple normal crossings and global-generation hypotheses, and comparing the canonical and algorithmic scattering diagrams.

Significance. If Theorem 4.73 is correct, it establishes a major, long-sought compatibility between two central constructions in mirror symmetry, and the extension to the relative Gross-Siebert locus in Chapter 5 is a substantial additional result. The paper is unusually careful about stating its assumptions, and it provides explicit, checkable computations in the elliptic curve case and in the P^3 K3 example. The strength of the work is, however, conditional: the proof of the main theorem relies on an extension of birational invariance of punctured log Gromov-Witten invariants beyond the precise statement quoted from [J], and Section 3.3.2 contains assertions rather than proofs for the removal of the simple normal crossings assumption. Because these points are load-bearing for the reduction to strongly admissible resolutions, the central claim is currently established only up to this external input. The paper does not provide machine-checked proofs or reproducible code, but its explicit computations and clear structural organization are valuable regardless.

major comments (4)
  1. [§1, Proposition 1.8] Proposition 1.8 is load-bearing for the reduction of Conjecture 1.7 to strongly admissible resolutions, yet its proof depends on an extension of [J, Corollary 1.6] that is not stated as a theorem anywhere. The sentence 'The proof does not use the fact that the radical of I is the maximal ideal' is an inference about the proof of an external result, not a proof of the needed statement. The needed statement concerns ideals I with radical J = P\K for the curves contracted by the logarithmic modification, which is exactly the non-maximal-radical case. Without a supplied proof, or a precise citation of a theorem covering this case, Theorem 4.73 establishes Conjecture 1.7 only for strongly admissible resolutions with maximal-radical base ideals.
  2. [§3.3.2] The removal of the simple normal crossings hypothesis is asserted rather than proved. The text states that the assumption 'can easily be removed' and that 'one can check that all the arguments of [GS8] work in this generalized setting', but no proof or detailed reference is given. This matters because admissible resolutions are allowed to have non-simple normal crossings D, so the comparison in Section 4.5 between the canonical scattering diagram and the algorithmic scattering diagram requires the non-snc version of the intrinsic mirror construction. The manuscript should either provide the promised argument or restrict Theorem 4.73 to the simple normal crossings case.
  3. [§3.3.4, Construction 3.69] The affine structure on the dual intersection complex in the general non-snc case is defined using [W, Theorem 4.1]. The hypotheses of that theorem are described only in passing, and it is not verified in the text that the log schemes X_{ρ_v} satisfy all of them in the situations needed for admissible resolutions. Since Construction 3.69 feeds directly into the definition of the canonical scattering diagram and hence into the main comparison, the precise content and applicability of [W, Theorem 4.1] should be stated and checked explicitly.
  4. [§4.5] The comparison of the canonical scattering diagram and the algorithmic scattering diagram for K3 surfaces relies on explicit punctured log Gromov-Witten computations taken from [G3] and [GHKS]. These computations are outsourced, and the text does not state exactly which results are used or verify that the hypotheses of those results are satisfied for the non-snc admissible resolutions introduced earlier. Since this comparison is the technical core of Theorem 4.73, the manuscript should list the precise external statements used and confirm that they apply in the full generality required.
minor comments (4)
  1. [§1, paragraph on Batyrev degenerations] The phrase 'Baryrev degeneration' appears once where 'Batyrev degeneration' is clearly intended; please fix the typo.
  2. [§2.3, Proposition 2.6] The notation X_{η(t)}^{∆} is introduced just before the proposition, but the symbol η(t) is reused for a power series without explicitly saying that it is the same function as in the displayed equation; a one-line clarification would help.
  3. [§3.3.3, Construction 3.57] The proof of Proposition 3.63 invokes Steinitz's theorem via a PL-embedding of B into R^3, but B is only known to be a polyhedral manifold homeomorphic to a sphere; it would be helpful to state the precise version of Steinitz's theorem used and to note whether a triangulation step is required.
  4. [§3.4.1] The beginning of Section 3.4.1 is truncated in the displayed text after 'Restricting', and the section appears to be incomplete; please check that the final version contains the full subsection.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: intrinsic and toric degeneration mirrors are independently constructed and then compared.

full rationale

The central claim (Theorem 4.73 / Theorem A) is a comparison between two mirror constructions defined by independent recipes: the toric degeneration mirror is obtained from the algorithmic scattering diagram of [GS3] on the dual intersection complex of \bar{X}, while the intrinsic mirror is obtained from the resolved log CY degeneration X via punctured log Gromov-Witten invariants and the canonical scattering diagram of [GS7, GS8]. The proof in Sections 4.4 and 4.5 works by relating these two scattering diagrams, so the isomorphism asserted in Conjecture 1.7 is not built into either construction. The paper does rely on external results ([J, Corollary 1.6] for birational invariance of punctured invariants, and [G3]/[GHKS] for explicit scattering computations), and Proposition 1.8 explicitly extends [J] from ideals with maximal radical to ideals with radical J = P \setminus K, saying 'the proof does not use the fact that the radical of I is the maximal ideal'. This is a flagged inference and a potential correctness risk if [J] does not cover the non-snc or non-maximal-radical cases, but it is not circular: the target theorem is not used to justify the external input. Self-citations to the Gross-Siebert framework are present but load-bearing only as technical machinery, not as a substitute for the comparison being proved.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The paper introduces no speculative physical entities. It compares two established mirror constructions, so the ledger records the external theorems and construction choices on which the comparison rests: scattering diagram machinery, punctured invariant computations, and the finitely many free parameters (slab functions, gluing data) of the toric degeneration mirror.

free parameters (2)
  • Initial slab functions f_rho (and constants a_{rho,i} for K3 surfaces) = unspecified elements of the ground field k
    Section 3.2.4 (3.25) and Proposition 3.33 (3.26). The toric degeneration mirror depends on these choices when the dual intersection complex is not simple; Theorem B varies them as free parameters.
  • Gluing data s for the central fibre of the toric degeneration = unspecified
    Section 5.4 and Assumption 3.37. The universal toric degeneration mirror of [GHS, Theorem A.4.2] is parameterized by gluing data; Theorem B extends the correspondence by varying them.
assumptions (5)
  • standard math Existence and uniqueness of the algorithmic scattering diagram (Theorem 3.35 of [GS3])
    Defines the toric degeneration mirror; the uniqueness up to combinatorial equivalence is used in Section 4.5 to identify the algorithmic and canonical diagrams.
  • standard math Consistency of the canonical scattering diagram (Theorem 3.90 of [GS8])
    The intrinsic mirror is constructed from this consistent scattering diagram (Section 3.3.8); the comparison relies on this being a well-defined invariant of the degeneration.
  • domain assumption Birational invariance of punctured log Gromov-Witten invariants ([J, Corollary 1.6])
    Used in Proposition 1.8 and Section 3.3.2; if false, the reduction to strongly admissible resolutions and the removal of global generation fail.
  • domain assumption Explicit punctured Gromov-Witten computations from [G3] and [GHKS]
    Section 4.4-4.5 uses their computed scattering diagrams for K3 degenerations as a black box; they are not reproduced here.
  • domain assumption The input degeneration is special (Definition 3.39), has smooth generic fibre, and the constructed resolution is minimal log CY satisfying Assumption 3.48
    These are the hypotheses of Conjecture 1.7 and Theorem A; the paper builds these resolutions but the notion of 'special' is itself a technical restriction on the local models.

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Pith. "Pith review of Mirrors to toric degenerations via intrinsic mirror symmetry." pith.science (2026). https://pith.science/paper/3AMPYQ4T

@misc{pith2026260807381,
  author       = {Pith},
  title        = {Pith review of: Mirrors to toric degenerations via intrinsic mirror symmetry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3AMPYQ4T}},
  note         = {Machine review of arXiv:2608.07381}
}
abstract

We explore the connection between two mirror constructions in Gross-Siebert mirror symmetry: toric degeneration mirror symmetry (arXiv:1212.4220, arXiv:math/0309070, arXiv:0709.2290, arXiv:math/0703822) and intrinsic mirror symmetry (arXiv:1909.07649, arXiv:2105.02502). After briefly exploring the case of degenerations of elliptic curves, we show that the Gross-Siebert mirror construction for minimal relative log Calabi-Yau degenerations generalizes that for divisorial toric degenerations $\bar{\mathfrak{X}} \to \mathcal{S}$ of K3-s that have a smooth generic fibre. We achieve this by constructing a resolution of $\bar{\mathfrak{X}} \to \mathcal{S}$ to a relative minimal log Calabi-Yau degeneration $\mathfrak{X} \to \mathcal{S}$ and comparing the algorithmic scattering diagram $\bar{\mathfrak{D}}$ giving rise to the toric degeneration mirror $\check{\bar{\mathfrak{X}}}$ and the canonical scattering diagram $\mathfrak{D}$ giving rise to the intrinsic mirror $\check{\mathfrak{X}}$. Moreover, we vastly expand the construction and obtain a correspondence between the restriction of the intrinsic mirror to the (numerical) minimal relative Gross-Siebert locus and the universal toric degeneration mirror. We also discuss generalizing the results to higher dimensions. In particular, we construct log smooth resolutions for a natural family of toric degenerations of Calabi-Yau threefolds.

Figures

Figures reproduced from arXiv: 2608.07381 by the authors.

Figure 2.1
Figure 2.1. Reflexive pairs p∆1, ∆˚ 1 q and p∆2, ∆˚ 2 q. 20The group of affine unimodular transformations consisting of translations by an integer vector and the linear transformations in GLp2, Zq [PITH_FULL_IMAGE:figures/full_fig_p020_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Equations for the toric degenerations X∆ in the Cox coordinates. 2.2. The mirror to a degeneration of elliptic curves. We want to compute the toric degeneration mirrors Xˇ TD and the intrinsic mirrors Xˇ IMS to the Batyrev degenerations X∆ of [PITH_FULL_IMAGE:figures/full_fig_p023_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. The central fibres of a degeneration of elliptic curves and its mirror. We shall now explain how to construct the toric degeneration mirror Xˇ TD and the intrinsic mirror Xˇ IMS to X Ñ S. We can make an explicit computation because the dual intersection complex B of X Ñ S has an affine structure with no singularities. In this situation, the recipes of [GS3] and [GS8] for constructing mirrors reduce to a simple rule.… view at source ↗
Figures from the paper (32 more)
Figure 2.4
Figure 2.4. Figure 2.4: The tropical curve corresponding to the product ϑ˜ ´1,1 ¨ ϑ˜ 2,1 “ ϑ˜ 1,2t D˜ρ0 when d ˚ i “ 2 for 0 ď i ď m ´ 1. Note that all the D˜ ρi with i “ řk i“0 d ˚ i ` jd˚ get identified with Dk under the quotient map Xˆ Σ Ñ Xˆ Σ{Z – X. Similarly, the theta functions ϑ˜ p …
Figure 2.5
Figure 2.5. Figure 2.5: Equations for the toric degeneration mirrors Xˇ ∆,TD. 2.3. Comparing the mirror families. We first note that Conjecture 1.7 tau￾tologically holds for toric degenerations of elliptic curves. Proposition 2.4. Let X Ñ S be a toric degeneration of elliptic curves with po…
Figure 3.1
Figure 3.1. Figure 3.1: Extensions of Conjecture 1.7. Finally, in Section 5.4, we study the fibres of the extended intrinsic mirror Xˇ Ñ Spec krt Eρ,k sJNE ` X¯ 0 ˘ K over points with arbitrary choices of ś4 k“1 t Eρ,k P k ˆ for ρ P P¯r1s (such that this system of equations is consistent67)…
Figure 4.2
Figure 4.2. Figure 4.2: Two blowups of an ODP. x “ t “ 0 y “ t “ 0 x “ wρ “ 0 y “ wρ “ 0 Blowup of x “ t “ 0 x “ t “ 0 y “ t “ 0 x “ wρ “ 0 y “ wρ “ 0 ODP singularity x “ t “ 0 y “ t “ 0 x “ wρ “ 0 y “ wρ “ 0 Blowup of y “ t “ 0 [PITH_FULL_IMAGE:figures/full_fig_p112_4_2.png]
Figure 4.1
Figure 4.1. Figure 4.1: Two resolutions of an ODP. The exceptional locus of the blowup is a single curve E meeting tx “ y “ t “ 0u at one point and contained in the divisor that was blown up. See [PITH_FULL_IMAGE:figures/full_fig_p112_4_1.png]
Figure 4.3
Figure 4.3. Figure 4.3: Blowing up the divisor D¯ 1. The number of singular￾ities on every X¯ ρ “ D¯ 1 X D¯ j , j “ 1, 2, 3 corresponds to rρ. the adjacent divisors at one point. Every exceptional curve arises as the curve E in Section 4.1.1 using the local model at the corresponding singul…
Figure 4.4
Figure 4.4. Figure 4.4: Transformation of ` B, ¯ P¯ ˘ corresponding to blowing up D¯ 1. above. Under the assumption that all the σ P P¯max are standard triangles, we have pB2 ,P2 q – ` B, ¯ P¯ ˘ as polyhedral manifolds and pB2 ,P2 q has the struc￾ture of an affine manifold with singularitie…
Figure 4.5
Figure 4.5. Figure 4.5: Types of σ P P¯max in Section 4.2. The motivation for restricting the types of σ P P¯max to [PITH_FULL_IMAGE:figures/full_fig_p120_4_5.png]
Figure 4.6
Figure 4.6. Figure 4.6: Subdivisions corresponding to divisor blowups [PITH_FULL_IMAGE:figures/full_fig_p122_4_6.png]
Figure 4.7
Figure 4.7. Figure 4.7: Subdivisions of more general σ P P¯max . We expect that the types of σ P P¯max in [PITH_FULL_IMAGE:figures/full_fig_p122_4_7.png]
Figure 4.8
Figure 4.8. Figure 4.8: Resolution of txy “ t 4 u Ď Spec krx, y, wρsJtK. this procedure l ´ 1 times (with choices of divisors to blow up at every step) produces a simple normal crossings degeneration. Notation 4.10. We will follow the convention that for every successive blowup, the strict …
Figure 4.9
Figure 4.9. Figure 4.9: Blowing up txy “ t 4 u Ď Spec krx, y, wρsJtK. divisors tx “ t “ 0u or tx 1 “ t “ 0u at each step l times (using Notation 4.10). Indeed, suppose that l ą 1 (otherwise, we are in the case of an ODP singularity of Section 4.1.1). Blowing up tx “ t “ 0u (without loss of …
Figure 4.10
Figure 4.10. Figure 4.10: Resolution of txy “ t 4wρu Ď Spec krx, y, wρsJtK. The exceptional locus after all the l blowups is a chain of l ´ 1 new divisors D1, . . . , Dl´1 that separate the strict transforms of the original divisors tx “ t “ 0u and ty “ t “ 0u, along with an exceptional curv…
Figure 4.11
Figure 4.11. Figure 4.11: Blowing up txy “ t 4wρu Ď Spec krx, y, wρsJtK. Remark 4.11. Note that performing a similar construction in the case of the more general divisorial singularity with local model ␣ xy “ t lw k ρ ( Ď Spec krx, y, wρsJtK for k ě 2 (see Observation 3.43(1)) is not possibl…
Figure 4.12
Figure 4.12. Figure 4.12: Blowing up the divisor D¯ 1. Here lρ13 “ 1, lρ12 “ 3, and lρ15 “ lρ16 “ 2 (one can only see that lρ12 , lρ15 , lρ16 ą 1 from this sketch). We explain the effect of blowing up D¯ i on the dual intersection complex ` B, ¯ P¯ ˘ . As usual, we define the dual intersecti…
Figure 4.13
Figure 4.13. Figure 4.13: Transformation of ` B, ¯ P¯ ˘ corresponding to blowing up D¯ 1. affine structure on Wv¯i zv¯i . We also have an intrinsic affine structure on Wv¯ρ,1 zv¯ρ,1 for all ρ P P¯r1s with X¯ ρ Ď D¯ i and lρ ą 1. By Remark 3.70(3), it extends to the whole Wv¯ρ,1 since D¯ ρ,1 …
Figure 4.14
Figure 4.14. Figure 4.14: Blowing up D¯ 1 again and the corresponding trans￾formation of the dual intersection complex. for the exceptional curves of the P 1 -bundles D¯ij p (and their strict trans￾forms after further blowups). We may also use alternative notations Dρ,p and Fρ,p. We write v …
Figure 4.15
Figure 4.15. Figure 4.15: Blowing up D¯ 1 (once again) and the corresponding transformation of the dual intersection complex. Now, to resolve the singularity on D¯ 12 2 X D¯ 1, we can blow up either D¯ 12 2 or D¯ 1. Either way, this resolves the ODP singularity on D¯ 12 2 X D¯ 1, producing a…
Figure 4.16
Figure 4.16. Figure 4.16: DJ , D0, ΦpD0q 1 , and D¯ 0 in the neighbourhood of xv1, v5y of [PITH_FULL_IMAGE:figures/full_fig_p173_4_16.png]
Figure 6.1
Figure 6.1. Figure 6.1: Resolution of the local model X˜¯ τ,x for τ a standard triangle [PITH_FULL_IMAGE:figures/full_fig_p215_6_1.png]
Figure 6.2
Figure 6.2. Figure 6.2: Blowing up the local model X˜¯ τ,x. Similarly to the analysis of Section 4.1, blowing up a component D¯ v, v P P¯r0s of the central fibre X¯ 0 of X¯ gives a small partial resolution X 1 Ñ X¯ that is trivial in the local models for the codimension 3 strata X¯ σ, σ P P…
Figure 6.3
Figure 6.3. Figure 6.3: Transformation of a cell τ P P¯r2s under blowing up components. Blow up all the irreducible components of the central fibre X¯ 0 of X¯ Ñ S in any order. By the analysis above, this gives a resolution π : X Ñ X¯ to a log smooth degeneration X Ñ S. Here the log structu…
Figure 6.4
Figure 6.4. Figure 6.4: Types of singular loci of τ P P¯r2s . We will now construct a subdivision of τ (or some rescaling of τ ) and an induced subdivision of ∆τ that resolves the local model X˜¯ τ,x. We consider a special case first. Definition 6.5. Let τ P P¯r2s be a cell of codimension 1…
Figure 6.5
Figure 6.5. Figure 6.5: Subdivisions of easily decomposable τ P P¯r2s . Notation 6.7. We confuse ∆τ,1 and the polyhedra of [PITH_FULL_IMAGE:figures/full_fig_p220_6_5.png]
Figure 6.6
Figure 6.6. Figure 6.6: Refined subdivisions of easily decomposable τ P P¯r2s . The subdivisions of [PITH_FULL_IMAGE:figures/full_fig_p221_6_6.png]
Figure 6.7
Figure 6.7. Figure 6.7: Resolution of the local model X˜¯ σ,x for σ a standard square. In [PITH_FULL_IMAGE:figures/full_fig_p222_6_7.png]
Figure 6.8
Figure 6.8. Figure 6.8: Transformation of σ corresponding to [PITH_FULL_IMAGE:figures/full_fig_p222_6_8.png]
Figure 6.9
Figure 6.9. Figure 6.9: Final subdivisions and singular loci of easily decom￾posable τ P P¯r2s . 6.1.3. Resolutions of the local models in general. Let x be a singular point con￾tained in the minimal stratum X¯ τ for τ P Pr2s a codimension 1 cell (we no longer assume that τ is easily decomp…
Figure 6.10
Figure 6.10. Figure 6.10: Subdivisions of general τ P P¯r2s . of [PITH_FULL_IMAGE:figures/full_fig_p225_6_10.png]
Figure 6.11
Figure 6.11. Figure 6.11: Subdivision of kσ. The subdivision of [PITH_FULL_IMAGE:figures/full_fig_p226_6_11.png]
Figure 6.12
Figure 6.12. Figure 6.12: Subdivision of kσ1. Note that all the ηj , 1 ď j ď q in [PITH_FULL_IMAGE:figures/full_fig_p226_6_12.png]
Figure 6.13
Figure 6.13. Figure 6.13: The family of tropical curves in pB,Pq correspond￾ing to τa1,a2 . By the discussion above and Construction 3.89 of the canonical scattering di￾agram, Conjecture 6.24 implies Conjecture 6.23 in the case that n “ 3 and all the σ P P¯max are standard simplices. The fac…

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