1 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:
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
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
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
Bocklandt, R., Craw, A., Vélez, A.Q.: Geometric Reid’s recipe for dimer models. Math. Ann. 361(3–4), 689–723 (2015)
Ishii, A., Ueda, K.: Dimer models and Crepant resolutions. Hokkaido Math. J. 45(1), 1–42 (2016)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
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
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-020-02127-w