Generic slicing identifies flop stability walls with restricted ADE walls

Since $r_0=1$, any independent curve coordinates $({\theta }_i{)}_{i\in I}$ determine the unique weight ${\theta }_0=-{\sum }_{i\in I}{r}_i{\theta }_i$. Extending these coordinates by zero on $J$ gives a linear bijection ${\mathrm{\Theta }}_r(\mathrm{\Lambda })\to L$, with inverse given by restriction to $I$ and that formula for ${\theta }_0$. The same construction applies to $A$ because taking the bundle dual preserves each summand rank. The evaluation convention in [F5] identifies these coordinates with the specified subspace of $E^{\ast }$.

Write $S=\mathbb{C}\mathrm{⟦}{z}_1,z_2,z_3,z_4\mathrm{⟧}$, $\mathfrak{m}=(z_1,z_2,z_3,z_4)$ and $R=S\mathrm{/}(F)$ as in the setting. This is an isolated complete intersection of dimension three over the infinite field $\mathbb{C}$, so [F4] applies to the proper principal ideal $(F)$. Choose a determinacy order $k$ at least the order of $F$, and let $P$ be its degree-$k$ polynomial truncation. Then $F-P\in {\mathfrak{m}}^{k+1}$ and $P\mathrm{\ne }0$, so contact determinacy gives $R\cong S\mathrm{/}(P)$ as complete-local $\mathbb{C}$-algebras. For the finitely generated algebra $B=\mathbb{C}[z_1,z_2,z_3,z_4]\mathrm{/}(P)$ and its origin maximal ideal $\mathfrak{n}$, the finite jets of $B_{\mathfrak{n}}$ are $S\mathrm{/}((P)+{\mathfrak{m}}^q)$; their inverse limit is $S\mathrm{/}(P)$. Thus $R\cong \widehat{B_{\mathfrak{n}}}$. The ADE slice also has a polynomial presentation by its stated Du Val normal form. Transporting the fixed morphism, slice and bundles along these complete-base isomorphisms preserves their properties and rank-vector stability. Hence the completion convention of Wemyss §1.6 holds for the required bases, with no restriction on the formal series originally chosen.

Bundle duality sends an endomorphism $a$ of $\mathcal{T}$ to $a^{\vee }$ on $\mathcal{V}$, with $(ab{)}^{\vee }=b^{\vee }{a}^{\vee }$, and double duality is its inverse. Thus $\mathrm{\Lambda }\cong A^{\mathrm{o}\mathrm{p}}$, preserving each labelled summand projection. For a finite-dimensional right $A$-module $Q$, its vector-space dual $DQ={\mathrm{Hom}}_{\mathbb{C}}(Q,\mathbb{C})$ is a right $\mathrm{\Lambda }$-module of the same dimension vector. Submodules $U\subseteq DQ$ correspond by annihilators to quotients $Q\mathrm{/}N$, and $\dim U=\dim (Q\mathrm{/}N)$ vertex by vertex. If $\dim Q=r$ and $\theta (r)=0$, then $(-\theta )(U)=\theta (N)$. Nonzero proper submodules correspond on the two sides. Hence $Q$ is $\theta$-semistable, respectively stable, exactly when $DQ$ is $(-\theta )$-semistable, respectively stable. Applying double vector-space duality gives the converse and transports the chamber and wall structures by $\theta \mapsto -\theta$.

The fixed morphism has smooth source and is a complete-local crepant flopping contraction, and $A$ is exactly the algebra of the basic bundle $\mathcal{T}$ in [F2] after translating its module and multiplication conventions. Step 1.2 verifies the additional completion hypothesis in [F2]. Apply the imported slice and surface wall theorems to this algebra with its rank vector. By [F3] and the root-system conventions [F5], they give precisely the proper restricted ADE root hyperplanes in the evaluation coordinates of step 1.1. Although the GIT source uses rational weights, for fixed $r$ there are only finitely many possible submodule dimension vectors $\beta$, with $0\le {\beta }_i\le r_i$. The signs and zeros of the finitely many integral forms $\theta \cdot \beta$ therefore determine semistability and stability of every such module. Every nonempty real sign stratum contains rational points: its equalities define a rational linear subspace, and rational points in it approximate any point satisfying its finitely many strict inequalities. Thus the rational chamber description extends to the real stability space without additional walls.

Negation fixes $L$ and every central restricted root hyperplane as a set, so the duality comparison in step 2.1 leaves the wall collection from step 2.2 unchanged. This gives the stated walls for $\mathrm{\Lambda }$ in the chosen surviving-curve coordinates. The identification and simpliciality in [F3] now match this arrangement with ${\mathcal{H}}_{\mathrm{\Lambda }}$ for flop and mutation functors. Since the imported construction is a restriction arrangement, it supplies no general assertion that this arrangement is Coxeter. The completion description of $R$ is proved in step 1.2.