Correction to: Geometric Reid’s recipe for dimer models

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,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 ${\overline{C}}\cap S_\rho ^\perp$.