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cD4 geometric occurrence and core-coset context

PDF p. 27: §7 paragraph after the Coxeter diagram

Statement

For the $D_{4}$ diagram with central vertex $2$ and $J = {2}$ , the restricted arrangement is linearly identical to Hirano–Wemyss Example 2.2. This is the seven-hyperplane arrangement occurring for $c D_{4}$ singularities with three curves meeting at one point in the threefold flopping setting. Here $c D_{4}$ means a complex complete-local three-dimensional hypersurface singularity whose sufficiently general hyperplane section is formally isomorphic to the Du Val surface $Spec C ⟦ a, b, c ⟧ / (a^{2} + b^{2} c + c^{3})$ . The three curves are components of the reduced closed fibre of a chosen crepant partial resolution, and its associated arrangement is defined by the generic-slice construction below. The geometric occurrence is an independent literature input, not a derivation of the combinatorics.

The same central-vertex parabolic datum occurs in Ko §8.2 on core double cosets: Ko's central generator $c$ corresponds to $s_{2} = t$ here. Ko Examples 1.9 and 4.14 also use the central numerical label $2$ , but their parabolic subset ${1, 3}$ is different from $J$ .

Statement references

D4 diagram and generator conventions

Seven relative D4 hyperplanes and thirty-two chambers

Core and atomic double cosets

Complete-local smooth threefold flops and their slice data

Facts & Assumptions

[F1]

By D4 diagram and generator conventions, the central generator is $s_{2} = t$ , with leaves $s_{1} = s, s_{3} = u, s_{4} = v$ . By Seven relative D4 hyperplanes and thirty-two chambers, on $Θ_{J} = {f : f (α_{2}) = 0}$ the coordinates $X = f (α_{1}), Y = f (α_{3}), Z = f (α_{4})$ give exactly the seven nonzero forms $X, Y, Z, X + Y, X + Z, Y + Z, X + Y + Z$ , up to proportionality. Their kernels are all the relative hyperplanes.

[F2]

The cited external inputs provide the geometric occurrence, Wemyss's seven equations, and Ko's stated parabolic data. The terminology core and atomic double cosets is as in Core and atomic double cosets. No geometric input is used to establish [F1].

[F3]

Use the projectivity, partial-resolution, crepancy and Du Val surface terminology of Complete-local smooth threefold flops and their slice data. In particular, a crepant partial resolution $f : X \to Spec R$ here is projective and birational, has normal source, is an isomorphism over the nonsingular locus, and satisfies $ω_{X} ≅ f^{*} ω_{R}$ compatibly there. A general hyperplane section is $R / (g)$ for a sufficiently general complex linear combination $g$ of the four ambient formal coordinates. In the smooth case, a contraction with exceptional locus consisting only of curves is the flopping-contraction case. Only the stated terminology of the setting definition is used here; its additional isolated-base hypothesis is not imposed on the toric example in [F5].

[F4]

The generic-slice input from Wemyss survey §3.3 gives $Y = X \times_{Spec R} Spec (R / (g))$ and a factorization $Y_{\min} \overset{h}{\to} Y \to Spec (R / (g))$ , with $Y_{\min}$ the minimal Du Val surface resolution. Its exceptional curves have finite ADE dual graph $Δ$ ; let $J_{f}$ be the vertices whose curves $h$ contracts. The surviving curves correspond to the reduced irreducible curves in $f^{- 1} (0)$ . By the survey §3.1 construction, the arrangement attached to this slice datum is the collection of distinct proper root-hyperplane intersections with $Θ_{J_{f}} = {ϑ \in R^{Δ} : ϑ_{j} = 0 (j \in J_{f})}$ . For a root $α = \sum_{i} c_{i} α_{i}$ , the root hyperplane is ${ϑ : \sum_{i} c_{i} ϑ_{i} = 0}$ . Thus the arrangement is attached to the chosen resolution and its slice, rather than to the threefold equation alone.

[F5]

A specific witness is the crepant resolution $f_{0} : X_{0} \to Spec R_{0}$ in Wemyss Examples 3.4 and 7.6, where $R_{0} = C ⟦ u, x, y, z ⟧ / (u^{2} - x y z)$ . Its reduced closed fibre is $C_{1} \cup C_{2} \cup C_{3}$ , with $C_{i} ≅ P^{1}$ and a common intersection point. Remark 7.5 identifies its chamber arrangement with Figure 6. Example 7.6, Step 1, supplies the contraction of each $C_{i}$ individually as a flopping contraction. These individual contractions provide the flopping context; the full morphism $f_{0}$ is not assumed small. The resolution, fibre, chamber and individual-flop assertions are imported, not proved here.

External input

Hirano–Wemyss Example 2.2 displays the seven equations $ϑ_{1} = 0$ , $ϑ_{2} = 0$ , $ϑ_{3} = 0$ , $ϑ_{1} + ϑ_{2} = 0$ , $ϑ_{1} + ϑ_{3} = 0$ , $ϑ_{2} + ϑ_{3} = 0$ , $ϑ_{1} + ϑ_{2} + ϑ_{3} = 0$ , and explicitly records their occurrence for $c D_{4}$ singularities with three curves meeting at a point.

Wemyss Example 7.4 and Figure 6 display the walls with forms $a, b, c, a + c, b + c, a + b + c, a + b + 2 c$ . Remark 7.5 identifies $R = C ⟦ u, x, y, z ⟧ / (u^{2} - x y z)$ as a $c D_{4}$ example with this chamber structure. Examples 3.4 and 7.6 specify the crepant resolution and its closed fibre in [F5]; Example 7.6, Step 1, states that each of its three curves is individually floppable.

Ko §8.2 uses type $D_{4}$ , central generator $c$ , and the core-coset collection indexed by ${c}$ . Examples 1.9 and 4.14 use central vertex $2$ and atomic cosets at ${1, 3}$ .

Wemyss survey Definition 1.1 and §2.1 supply the cDV and crepancy conventions. Sections 3.1 and 3.3 supply the generic slice, the minimal-resolution factorization, the surviving-curve correspondence and the attached finite restriction arrangement in [F4], for a chosen crepant partial resolution of a complete-local complex cDV singularity. These geometric statements are external inputs; their proofs are not reconstructed here.

Verification

Given: The central-vertex $D_{4}$ data, the completed restricted-arrangement calculation [F1], and the independent source statements [F2].

Goal: Identify the arrangements and numbering and substantiate the contextual occurrence.

1.1F1F2

Set $(ϑ_{1}, ϑ_{2}, ϑ_{3}) = (X, Y, Z)$ . This linear bijection sends the exhaustive seven forms in [F1] to precisely the seven displayed forms of Hirano–Wemyss Example 2.2. In particular no additional hyperplane or omitted hyperplane is concealed by the comparison.

2.1F1F2step 1.1

Wemyss Figure 6 uses a different chamber's coordinate system. Put $(a, b, c) = (X + Z, Y + Z, - Z)$ , an invertible change with inverse $(X, Y, Z) = (a + c, b + c, - c)$ . The seven forms in that figure become $X + Z, Y + Z, - Z, X, Y, X + Y + Z, X + Y$ , respectively. Their kernels are exactly those in step 1.1, since a form and its negative have the same kernel. Thus the coefficient $2$ in that figure does not produce a different arrangement. This identifies hyperplanes, without asserting that the displayed positive chambers or individual geometric curve labels agree.

3.1F1F2F3F4F5step 1.1step 2.1∎

Hirano–Wemyss's explicit occurrence statement, together with Wemyss Remark 7.5 and Example 7.6, supplies the claimed geometric context for this arrangement, with its singularity, fibre and attached-arrangement meanings fixed in [F3]–[F5]. For Ko §8.2 choose the diagram isomorphism $(c, t, u, v)_{K o} \mapsto (s_{2}, s_{1}, s_{3}, s_{4})$ ; it preserves all three central-to-leaf edges and sends ${c}$ to ${s_{2}}$ . Thus it is the same central-vertex parabolic datum. Ko's numerically labelled Examples 1.9 and 4.14 corroborate the central label but concern a different subset, as stated.

Source notes

The original article is not a mathematical input to this identification. The root restriction is supplied by the completed local calculation; the geometric existence is imported from the independent sources with the reading limits recorded above. The coordinate change compares unlabelled hyperplanes, not geometric marking data or derived equivalences. No original-article defect is asserted.

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External dependencies

Yuki Hirano and Michael Wemyss, Faithful actions from hyperplane arrangements · Example 2.2, preprint PDF p. 4

The arrangement in R^3 is defined by the seven nonempty subset sums of coordinates vanishing, has 32 chambers, and occurs for cD4 singularities with three curves meeting at a point.

Hypotheses: The displayed real three-dimensional arrangement in Example 2.2; the geometric assertion is an occurrence statement, not a universal assertion for all cD4 singularities.

Reading limits: Read Example 2.2, its seven equations and the preceding arrangement definitions in the online PDF text in this adjudication. The occurrence is imported; neither its geometric proof nor the CS example nor the faithfulness theorem was reconstructed. No rendered-source inspection is claimed for this adjudication.

Michael Wemyss, Flops and clusters in the homological minimal model programme · Example 3.4, printed p.20; §7.1 opening, p.47; Example 7.4, pp.48–49 and Figure 6; Remark 7.5 and Example 7.6, pp.49–50, through Step 1

Example 3.4 specifies the resolution of R0=C[[u,x,y,z]]/(u^2-xyz) using G-Hilb(C^3) for the quotient by Z2 x Z2. Remark 7.5 identifies its cD4 chamber structure with Figure 6, whose wall forms are a,b,c,a+c,b+c,a+b+c,a+b+2c. Example 7.6 states that its closed fibre has three curves meeting at one point and that each curve is individually floppable.

Hypotheses: The particular complete-local toric cD4 singularity and crepant resolution of Examples 3.4 and 7.6, in the crepant partial-resolution scope stated at the start of §7.1. The individual curve contractions are distinguished from the full map to Spec R0; no isolated-base or smallness assumption is imposed on that full map.

Reading limits: Read the online PDF text of Example 3.4, the §7.1 opening, Example 7.4 and the seven Figure 6 equations, Remark 7.5, and Example 7.6 through Step 1. Import the resolution, fibre, individual-flop and chamber assertions; no reconstruction of G-Hilb, the quiver calculation, the cD4 classification or the underlying geometric proofs is claimed. Source diagram rendering was unavailable here.

Hankyung Ko, An atomic Coxeter presentation · §8.2 opening, PDF p. 28; Example 1.9, p. 6; Example 4.14, p. 16

Section 8.2 takes type D4 with central generator c and studies core cosets indexed by {c}. Examples 1.9 and 4.14 use central numerical label 2 and atomic relations at subset {1,3}.

Hypotheses: Finite Coxeter type D4, central generator c in §8.2; the examples at {1,3} are different parabolic subsets from {c}.

Reading limits: Read the opening and displayed formulas of §8.2 and Examples 1.9 and 4.14 in the online v3 PDF text in this adjudication. Use only the diagram labels and parabolic subsets, not the later coset enumeration or the atomic presentation theorem. No rendered-source inspection is claimed.

Michael Wemyss, A lockdown survey on cDV singularities · Definition 1.1, printed p.1; §2.1, p.9; §2.3, pp.12–13; §3.1, pp.13–14; §3.3, p.15

A complete-local complex cDV ring is a hypersurface deformation of a Du Val surface. Crepancy means pullback equality of canonical sheaves. For a crepant partial resolution of a cDV singularity, a generic section gives a partial resolution of an ADE surface, dominated by its minimal resolution. The curves contracted by the latter map specify J in the ADE diagram, and the number of surviving curves equals the number above the threefold origin. The associated finite arrangement is the restriction of the ADE root arrangement to the coordinate subspace with coordinates in J zero.

Hypotheses: A complete-local complex cDV singularity and a chosen projective crepant partial resolution. The generic section is chosen as in §3.3. This construction permits nonisolated bases; a small smooth contraction is the special flopping case. Only the finite arrangement is used.

Reading limits: Read Definition 1.1, the crepancy and resolution definitions at the start of §2.1, the reduced-fibre paragraph of §2.3, and the finite-arrangement construction in §3.1 and generic-slice construction in §3.3 in the online PDF text. Import generic slicing and its curve correspondence without reconstructing their proofs. No affine-arrangement or Pinkham movable-cone theorem is used.

Sources

Yuki Hirano and Michael Wemyss, Faithful actions from hyperplane arrangements · Example 2.2, preprint PDF p. 4
Michael Wemyss, Flops and clusters in the homological minimal model programme · Example 3.4, printed p.20; §7.1 opening, p.47; Example 7.4, pp.48–49 and Figure 6; Remark 7.5 and Example 7.6, pp.49–50, through Step 1
Hankyung Ko, An atomic Coxeter presentation · §8.2 opening, PDF p. 28; Example 1.9, p. 6; Example 4.14, p. 16
Michael Wemyss, A lockdown survey on cDV singularities · Definition 1.1, printed p.1; §2.1, p.9; §2.3, pp.12–13; §3.1, pp.13–14; §3.3, p.15