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)

    MathSciNet  Article  Google Scholar 

  2. 2.

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

    MathSciNet  Article  Google Scholar 

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Bocklandt, R., Craw, A. & Quintero Vélez, A. Correction to: Geometric Reid’s recipe for dimer models. Math. Ann. 380, 911–913 (2021). https://doi.org/10.1007/s00208-020-02127-w

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