# Correction to: Geometric Reid’s recipe for dimer models

The Original Article was published on 21 August 2014

## Correction to: Mathematische Annalen https://doi.org/10.1007/s00208-014-1085-8

The main results of [1], especially Theorems 1.1, 1.4 and Corollary 1.2, are correct as written. However, the final sentence in the statement of Proposition 1.3 is false when the quiver $$\mathrm {Q}$$ contains a loop at a vertex $$i\in \mathrm {Q}_0$$. When this is the case, there exist points $$y\in {\mathcal {M}}_\vartheta$$ for which the corresponding A-module $$V_y$$ contains a submodule of dimension vector $$S_i$$ that is not isomorphic to $$S_i$$; note that any such $$V_y$$ is not nilpotent. This situation is very rare,Footnote 1 but it does occur.

### Example 1

For the action of type $$\frac{1}{2}(1,1,0)$$, let y be a generic point in the (noncompact) exceptional divisor in $$G{\text {-Hilb}}({\mathbb {C}}^3)$$, so $$y\not \in \tau ^{-1}(x_0)$$. The nonzero maps in the A-module $$V_y$$ are shown

where $$\rho _0$$ and $$\rho _1$$ are the trivial and nontrivial representations of $${\mathbb {Z}}/2$$ respectively. The submodule $$W_y\subset V_y$$ of dimension vector $$S_1$$ destabilises $$V_y$$ as $$\vartheta \in C$$ moves into the wall $${\overline{C}}\cap S_i^\perp$$ of the GIT chamber C, so y lies in the unstable locus of this wall. However, $$W_y\not \cong S_1$$, so $$S_i\not \subseteq {\text {soc}}(V_y)$$.

This example shows that even when $${\overline{C}}\cap S_i^\perp$$ is a wall of the chamber C, the unstable locus of the wall need not coincide with $$Z_i:=\{y\in Y \mid S_i\subseteq {\text {soc}}(V_y)\}$$. In such cases, the final sentence of Proposition 1.3 is false; that sentence should instead conclude that:

\begin{aligned}&... \text {the locus }Z_i\text { is }{} the intersection of \tau ^{-1}(x_0) with \nonumber \\&\qquad \text { the unstable locus of the wall } {\overline{C}}\cap S_i^\perp . \end{aligned}
(0.1)

Indeed, $$Z_i$$ is a subset of the unstable locus of the wall, but this inclusion is equality if and only if the unstable locus is contained in $$\tau ^{-1}(x_0)$$. Instead, for any point $$y\in \tau ^{-1}(x_0)$$ that lies in the unstable locus for the wall $${\overline{C}}\cap S_i^\perp$$, the destabilising submodule $$W_y\subset V_y$$ of dimension vector $$S_i$$ is necessarily isomorphic to $$S_i$$ since $$V_y$$ is nilpotent, giving $$y\in Z_i$$. This proves the equality (0.1).

The error leads to the omission of a case from Lemma 4.8. We now correct that statement:

### Lemma 2

(= Lemma 4.8) Every wall of the chamber C that is of form $${\overline{C}}\cap S_i^\perp$$ for some nonzero $$i\in \mathrm {Q}_0$$ is either of type $${\text {0}}$$, type $${\text {I}}$$, or it is a type $${\text {III}}$$ wall with unstable locus $${\mathbb {P}}^1\times {\mathbb {C}}$$. In particular, the support of $$H^0(\Psi (S_i))$$ is a single $$(-1,-1)$$-curve (in type $${\text {I}}$$), a single $$(0,-2)$$-curve (in type $${\text {III}}$$) or a connected union of compact torus-invariant divisors (in type $${\text {0}}$$).

### Proof

The only walls that are excluded here are type $${\text {III}}$$ walls for which the unstable locus is $${\mathbb {F}}_n$$ for some $$n\ge 0$$. Suppose that one such wall exists. The wall is of the form $${\overline{C}}\cap S_i^\perp$$, so Proposition 4.7 implies that $$\Psi (S_i) = L_i^{-1}\vert _{Z_i}$$. Since $${\mathbb {F}}_n\subseteq \tau ^{-1}(x_0)$$, the locus $$Z_i$$ coincides with the unstable locus $${\mathbb {F}}_n$$, so the support of $$\Psi (S_i)$$ is of dimension two. To obtain a contradiction, let $$\ell \subset Y$$ be the fibre of the contraction $${\mathbb {F}}_n\rightarrow {\mathbb {P}}^1$$ induced by the wall. For any $$z\in \ell$$, the sequence

(0.2)

is the $$\theta _0$$-destabilising sequence for $$V_z$$. In particular, the proof of Ishii–Ueda [2, Proposition 11.31] gives that $$\Psi (S_i) = {\mathcal {O}}_{\ell }(-1)$$, so the support of $$\Psi (S_i)$$ has dimension one, a contradiction. The second statement follows from (0.1) above, where in the type $${\text {III}}$$ case we compute $$Z_i$$ to be the intersection of $$\tau ^{-1}(x_0)$$ with the unstable locus $${\mathbb {P}}^1\times {\mathbb {C}}$$, i.e. $$Z_i$$ is the torus-invariant $$(0,-2)$$-curve in $${\mathbb {P}}^1\times {\mathbb {C}}$$. $$\square$$

The additional case of the type $${\text {III}}$$ wall in Lemma 2 should have been analysed in [1, Lemma 4.10, Proposition 4.11]. We now correct those omissions.

### Lemma 3

(= Lemma 4.10) Let $$\ell$$ be a $$(-1,-1)$$-curve or a $$(0,-2)$$-curve in Y that arises as the intersection of $$\tau ^{-1}(x_0)$$ with the unstable locus for a wall of the form $${\overline{C}}\cap S_i^{\perp }$$ for some nonzero $$i\in \mathrm {Q}_0$$ that is of type $${\text {I}}$$ or type $${\text {III}}$$ respectively. Then $$L_j\vert _{\ell }\cong {\mathscr {O}}_\ell$$ for all $$j\ne i$$ and $$L_i\vert _{\ell }\cong {\mathscr {O}}_\ell (1)$$.

### Proof

The proof from [1, Lemma 4.10] for a $$(-1,-1)$$-curve applies verbatim for a $$(0,-2)$$-curve, but the appropriate reference to the work of Ishii–Ueda in this latter case is [2, Lemma 11.32]. $$\square$$

### Proposition 4

(= Proposition 4.11) Let $$i\in \mathrm {Q}_0$$ be a nonzero vertex. If $$H^{0}(\Psi (S_i))\ne 0$$, then $$\Psi (S_i) \cong L_i^{-1}\vert _{Z_i}$$, where $$Z_i$$ is the intersection of $$\tau ^{-1}(x_0)$$ with the unstable locus for the wall $${\overline{C}}\cap S_i^\perp$$.

### Proof

The additional case from Lemma 3 shows that the support of $$H^{0}(\Psi (S_i))$$ can be a single $$(0,-2)$$-curve $$\ell _i$$ equal to the locus $$Z_i$$ for a type $${\text {III}}$$ wall $${\overline{C}}\cap S_i^\perp$$. The proof from [1, Proposition 4.11] in the case where $$\ell _i$$ is a $$(-1,-1)$$-curve applies verbatim, except that the required isomorphisms $$L_j\vert _{\ell _i}\cong {\mathscr {O}}_{\ell _i}$$ for all $$j\ne i$$ and $$L_i\vert _{\ell _i}\cong {\mathscr {O}}_{\ell _i}(1)$$ are obtained from Lemma 3. $$\square$$

The final correction is in [1, Proof of Theorem 1.1], where in describing the case $$H^{0}(\Psi (S_i))\ne 0$$, the locus $$Z_i$$ should equal the intersection of $$\tau ^{-1}(x_0)$$ with the unstable locus of the wall $${\overline{C}}\cap S_i^\perp$$. In particular, this locus $$Z_i$$ can be either a single $$(-1,-1)$$-curve, a single $$(0,-2)$$-curve or a connected union of compact torus-invariant divisors according to the type of the wall as in Lemma 2.

It remains to note that [1, Conjecture 1.5] should refer to the intersection of $$\tau ^{-1}(x_0)$$; in what follows, we take the determinant of $$L_\rho ^\vee$$ before restricting to $$Z_\rho$$ (this operation was omitted in [1]):

### Conjecture 5

(= Conjecture 1.5) The object $$\Psi (S_\rho )$$ is a pure sheaf in degree 0 if and only if $${\overline{C}}\cap S_\rho ^\perp$$ is a wall of the chamber C defining $$G{\text {-Hilb}}({\mathbb {C}}^3)$$, in which case $$\Psi (S_\rho )\cong \det (L_\rho ^{\vee })\vert _{Z_\rho }$$ where $$Z_\rho$$ is the intersection of $$\tau ^{-1}(x_0)$$ with the unstable locus of the wall $${\overline{C}}\cap S_\rho ^\perp$$.

## Notes

1. 1.

If $$\mathrm {Q}$$ has a loop at vertex $$i\in \mathrm {Q}_0$$, then the locus $$\tau ^{-1}(x_0)$$ is one-dimensional. Indeed, let $$n_1,\dots , n_k\in N$$ be the corners of the polygon P and write $$\Pi _1,\dots , \Pi _k$$ for the corresponding perfect matchings. Let $$m\in M$$ correspond to the loop $$\ell$$ in $$\mathrm {Q}$$ at vertex i. After reordering the corner perfect matchings if necessary, there exists $$1\le l\le k$$ such that $$\ell \in \Pi _j$$ if and only if $$1\le j\le l$$. Then $$\langle n_i,m\rangle =\deg _{\Pi _i} \ell =1$$ for $$1\le i\le l$$, whereas $$\langle n_j,m\rangle =0$$ for $$l+1\le j\le k$$. Choose a $${\mathbb {Z}}$$-basis of N such that the affine span of P is the plane $$\{(x,y,1)\in N\otimes {\mathbb {R}}\mid x, y\in {\mathbb {R}}\}$$. If we write $$m:=(u,v,w)\in M$$ in the dual coordinates, then the polygon P is sandwiched between the parallel lines $$ux+vy= -w$$ and $$ux+vy =-w+1$$ in the plane. Thus, P contains no internal lattice points, so $$\tau ^{-1}(x_0)$$ contains no surfaces. The proves the assertion.

## References

1. 1.

Bocklandt, R., Craw, A., Vélez, A.Q.: Geometric Reid’s recipe for dimer models. Math. Ann. 361(3–4), 689–723 (2015)

2. 2.

Ishii, A., Ueda, K.: Dimer models and Crepant resolutions. Hokkaido Math. J. 45(1), 1–42 (2016)

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Correspondence to Alastair Craw.

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