Multigraded linear series and recollement

Theorem 1.1

If the vector bundles \(E_1,\ldots , E_n\) are globally generated,  then there is a morphism \(f:Y\rightarrow {\mathcal {M}}(E)\) satisfying \(E_i=f^*(T_i)\) for \(0\le i\le n\) whose image is isomorphic to the image of the morphism \(\varphi _{\vert L\vert }:Y\rightarrow \vert L\vert \) to the linear series of \(L:=\bigotimes _{1\le i\le n} \det (E_i)^{\otimes j}\) for some \(j>0\).

Theorem 1.2

Proposition 1.3

The morphism \(g_C:{\mathcal {M}}(E)\rightarrow {\mathcal {M}}(E_C)\) is surjective iff for each \(x\in {\mathcal {M}}(E_C),\) the A-module \(j_!(N_x)\) admits a surjective map onto an A-module of dimension vector \(\varvec{\mathrm {v}}\).

Theorem 1.4

Suppose that for each \(x\in {\mathcal {M}}(E_C),\) the class \([\Psi (j_!(N_x))]\in K^{{\text {num}}}_c(Y)\) can be written as a positive combination of the classes of sheaves on Y. Then \(g_C\) is surjective.

Proposition 1.5

Remark 2.1

The ideal \(I\subset \mathbb {k}Q\) constructed in this way need not be admissible, and relations may even involve idempotents. For example, if \(E_i\cong E_j\) then the isomorphisms correspond to relations of the form \(aa^\prime -e_i, a^\prime a-e_j\in I\) for some \(a, a^\prime \in \mathbb {k}Q\).

Lemma 2.2

Proof

Proposition 2.3

Let \(E_1, \ldots , E_n\) be vector bundles on Y and set \(E_0={\mathcal {O}}_Y\). Then \(\bigoplus _{0\le i\le n} E_i\) is a flat family of 0-generated A-modules if and only if \(E_i\) is globally generated for all \(1\le i\le n\).

Proof

The bundle \(E:=\bigoplus _{0\le i\le n} E_i\) on Y is a flat family of A-modules of dimension vector \(\varvec{\mathrm {v}}=({{\mathrm{rk}}}(E_i))_{0\le i\le n}\), and for any closed point \(y\in Y\), the A-module structure in the fibre \(E_y\) of E over y is obtained by evaluating all homomorphisms between the bundles \(E_i\) (for \(0\le i\le n\)) at y. Choose \(\theta \in \Theta _{\varvec{\mathrm {v}}}^+\) satisfying \(\theta _i>0\) for \(i\ne 0\). Since \(\theta _0=-\sum _{1\le i\le n} v_i\theta _i\), an A-submodule W of \(E_y\) is destabilising if and only if \(\dim (W_0)=1\) and there exists \(i>0\) such that the sum of all maps from \(W_0\) to \(W_i\) determined by paths in the quiver from 0 to i is not surjective. In particular, \(E_y\) is \(\theta \)-unstable for some \(y\in Y\) if and only if there exists \(i>0\) such that \(E_i\) cannot be written as the quotient of \({\mathcal {O}}_Y^{\oplus k}\) for some \(k\in \mathbb {N}\); equivalently, \(E_y\) is \(\theta \)-stable for all \(y\in Y\) if and only if \(E_i\) is globally generated for \(1\le i\le n\). \(\square \)

Corollary 2.4

For any \(\theta \in \Theta _{\varvec{\mathrm {v}}}^+,\) the locally-free sheaf \(T_i\) on \({\mathcal {M}}(A,\varvec{\mathrm {v}},\theta )\) is globally generated for all \(0\le i\le n\).

Proof

The sheaf \(\bigoplus _{0\le i\le n} T_i\) is a flat family of \(\theta \)-stable A-modules of dimension vector \(\varvec{\mathrm {v}}\) on \({\mathcal {M}}(A,\varvec{\mathrm {v}},\theta )\). The result follows from Proposition 2.3 and the fact that \(T_0\cong {\mathcal {O}}_{{\mathcal {M}}(A,\varvec{\mathrm {v}},\theta )}\). \(\square \)

Definition 2.5

Theorem 2.6

Let \(E_1,\ldots , E_n\) be globally generated vector bundles on Y. There is a morphism \(f:Y\rightarrow {\mathcal {M}}(E)\) satisfying \(E_i=f^*(T_i)\) for \(1\le i\le n\) whose image is isomorphic to the image of the morphism \(\varphi _{\vert L\vert }:Y\rightarrow \vert L\vert \) to the linear series of \(L:=\bigotimes _{1\le i\le n} \det (E_i)^{\otimes j}\) for some \(j>0\).

Proof

Apply Proposition 2.3 and Lemma 2.2 \((\mathrm {ii})\) to the parameter \(\theta = j\theta ^\prime \in \Theta _{\varvec{\mathrm {v}}}^+\) from Definition 2.5, noting that \(E_0={\mathcal {O}}_Y\). \(\square \)

Example 2.7

(Linear series of higher rank) When Y is projective and \(E_1\) has rank r, then \({\mathcal {M}}(A,\varvec{\mathrm {v}},\theta )\) is isomorphic to the Grassmannian \({{\mathrm{Gr}}}(H^0(E_1),r)\) of rank r quotients of \(H^0(E_1)\). The ample bundle \({\mathcal {O}}(1)\) on \({{\mathrm{Gr}}}(H^0(E_1),r)\) is very ample, so we may take \(j=1\) in Definition 2.5 and Theorem 2.6. Therefore f coincides with the morphism \(\varphi _{\vert E_1\vert }\) to the linear series of higher rank that recovers \(E_1\) as the pullback of the tautological quotient bundle of rank r; see Mukai [29, Section 3]. When \(r=1\), this is the classical linear series of a basepoint-free line bundle.

Remark 2.8

Example 2.9

(Quiver flag varieties) Let Q be a finite, acyclic, connected quiver with a unique source denoted \(0\in Q_0\), and let \(\varvec{\mathrm {v}}=(v_i)\in \mathbb {N}^{Q_0}\) be a dimension vector satisfying \(v_0=1\). For \(\theta \in \Theta _{\varvec{\mathrm {v}}}^+\), the fine moduli space \({\mathcal {M}}(\mathbb {k}Q, \varvec{\mathrm {v}}, \theta )\) is called a quiver flag variety [15], and for \(i\in Q_0\), the tautological bundle \(T_i\) of rank \(v_i\) is globally generated by Corollary 2.4. Every such variety is an iterative Grassmann-bundle [15, Theorem 3.3], and \(T_i\) is simply the pullback of the tautological quotient bundle on one of the Grassmann bundles in the tower.

We may now apply Theorem 2.6 to the determinants of the tautological bundles. Indeed, \(\det (T_i)\) is globally generated for \(i\in Q_0\) and \(L=\bigotimes _{i\in Q_0} \det (T_i)\) is ample by [15, Lemma 3.7], so for \(E=\bigoplus _{i\in Q_0} \det (T_i)\), the morphism \(f:{\mathcal {M}}(T)\rightarrow {\mathcal {M}}(E)\) is a closed immersion.

Remark 2.10

Remark 3.1

This use of the term cornering comes from representation theory, where the basic example is cornering the matrix algebra \({{\mathrm{End}}}_{\mathbb {k}}(\mathbb {k}\oplus \mathbb {k}) \cong Mat _{2 \times 2}(\mathbb {k})\) by a nontrivial idempotent to produce a subalgebra of matrices that have nonzero entries only in one particular “corner”.

This is distinct from the construction of Ishii–Ueda [20] for dimer models, which is a process linking the removal of a corner in a lattice polygon to the universal localisation of certain arrows in a quiver in order to determine an open subset in an associated moduli space.

Lemma 3.2

For \(a_1,\ldots , a_n\ge 1,\) we have \({\mathcal {M}}(\bigoplus _{0\le i\le n}E_i)\cong {\mathcal {M}}(\bigoplus _{0\le i\le n}E_i^{\oplus a_i}).\)

Proof

The endomorphism algebra of \(E^\prime :=\bigoplus _{0\le i\le n}E_i\) is Morita equivalent to the endomorphism algebra of \(E:= \bigoplus _{0\le i\le n}E_i^{\oplus a_i}\). By increasing the multiplicities of the summands in the tautological bundle on \({\mathcal {M}}(E^\prime )\), we obtain a flat family \(\bigoplus _{0\le i\le n} T_i^{\oplus a_i}\) of 0-generated \({{\mathrm{End}}}(E)\)-modules of dimension vector \(\varvec{\mathrm {v}}= (a_i{{\mathrm{rk}}}(E_i))\), so there is a morphism \(f:{\mathcal {M}}(E^\prime )\rightarrow {\mathcal {M}}(E)\). For the other direction, we have \(E^\prime = E_C\) for some cornering subset C, so (3.1) defines a morphism \(f_C:{\mathcal {M}}(E)\rightarrow {\mathcal {M}}(E^\prime )\). Universality ensures that these morphisms are mutually inverse. \(\square \)

Proposition 3.3

Let \(E_1,\ldots , E_n\) be globally generated vector bundles on Y and set \(R:= \Gamma ({\mathcal {O}}_Y)\). There is a contravariant functor from \(\mathscr {C}\) to the category of R-schemes that sends a set C to the multigraded linear series \({\mathcal {M}}(E_C)\).

Proof

Given subsets \(C^\prime \subseteq C\subseteq \{0,1,\ldots , n\}\) that both contain \(\{0\}\), we first construct a morphism \(f_{C,C^\prime }:{\mathcal {M}}(E_C)\rightarrow {\mathcal {M}}(E_{C^\prime })\) that fits into a commutative diagram

Open image in new window

(3.2)

To avoid a proliferation of indices, we write \(T=\bigoplus _{i\in C}T_i\) and \(T^\prime =\bigoplus _{i\in C^\prime }T^\prime _i\) for the tautological bundles on \({\mathcal {M}}(E_C)\) and \({\mathcal {M}}(E_{C^\prime })\) respectively, where \({{\mathrm{rk}}}(T^\prime _i) = {{\mathrm{rk}}}(T_i)={{\mathrm{rk}}}(E_i)\) for all \(i\in C^\prime \). Since T is a flat family of 0-generated \(A_C\)-modules on \({\mathcal {M}}(E_C)\), it follows that \(\bigoplus _{i\in C^\prime }T_i\) is a flat family of 0-generated \(A_{C^\prime }\)-modules on \({\mathcal {M}}(E_C)\). The universal property of \({\mathcal {M}}(E_{C^\prime })\) as in Lemma 2.2 defines a morphism \(f_{C,C^\prime }:{\mathcal {M}}(E_C)\rightarrow {\mathcal {M}}(E_{C^\prime })\) satisfying \(f_{C,C^\prime }^*(T^\prime _i) = T_i\) for \(i\in C^\prime \). The morphism \(f_C\) is also universal, so

$$\begin{aligned} (f_{C,C^\prime }\circ f_C)^*(T_i^\prime ) = f_C^*(T_i) = E_i \end{aligned}$$

for all \(i\in C^\prime \). This property characterises \(f_{C^\prime }\), so diagram (3.2) commutes as required. That the assignment sending the inclusion \(C^\prime \hookrightarrow C\) to the morphism \(f_{C,C^\prime }\) respects composition follows similarly from the universal property of \(f_{C,C^\prime }\). Our assignment is therefore a functor.

It remains to prove that \(f_{C,C^\prime }\) is a morphism of R-schemes. For \(C^{\prime \prime }:=\{0\}\), we have \(A_{C^{\prime \prime }}={{\mathrm{End}}}(E_0) = R\) and hence \({\mathcal {M}}(E_{C^{\prime \prime }})=\mathrm{Spec}\,R\). The result follows by commutativity of the morphisms \(f_{C^{\prime },C^{\prime \prime }}\circ f_{C,C^\prime } = f_{C,C^{\prime \prime }}\) established above. \(\square \)

Definition 3.4

Example 3.5

For \(\mathbb {P}^2\) equipped with the tilting bundle \(E:={\mathcal {O}} \oplus {\mathcal {O}}(1) \oplus {\mathcal {O}}(2)\), the algebra \({{\mathrm{End}}}_{\mathbb {P}^2}(E)\) can be presented using the Beilinson quiver with relations:

Open image in new window

Consider the category \(\mathscr {C}\) from Proposition 3.3 viewed as a poset:

Open image in new window

It is easy to calculate that \({\mathcal {M}}(E) \cong \mathbb {P}^2\). Example 2.7 implies that \(g_{C_1}:{\mathcal {M}}(E) \rightarrow {\mathcal {M}}(E_{C_1})\) is an isomorphism, whereas \(g_{C_2}:{\mathcal {M}}(E) \rightarrow {\mathcal {M}}(E_{C_2}) \cong \mathbb {P}^5\) is the Veronese embedding \(\varphi _{\vert {\mathcal {O}}(2)\vert }\). For \(1\le i\le 3\), we have that \({\mathcal {M}}(E_{C_i}) \rightarrow {\mathcal {M}}(E_{C_0}) \cong \mathrm{Spec}\,\mathbb {k}\) is the structure morphism.

Lemma 3.6

Proof

Proposition 3.7

A closed point \(x \in {\mathcal {M}}(E_C)\) lies in the image of the cornering morphism \(g_C:{\mathcal {M}}(E) \rightarrow {\mathcal {M}}(E_C)\) if and only if the A-module \(j_!(N_x)\) admits a surjective map onto an A-module of dimension vector \(\varvec{\mathrm {v}}\).

Proof

For the opposite direction, let \(j_!(N_x)\rightarrow M\) be a surjective A-module homomorphism, where M has dimension vector \(\varvec{\mathrm {v}}\). Since \(N_x\) is 0-generated, Lemma 3.6 \((\mathrm {i})\) gives that \(j_!(N_x)\) is 0-generated and hence so is M. As such, M is a 0-generated A-module of dimension \(\varvec{\mathrm {v}}\), so M is \(M_y\) for some closed point \(y \in {\mathcal {M}}(E)\). Then \(N_x \cong j^*j_!(N_x) \cong j^*(M_y)\), giving \(g_C(y) =x\). \(\square \)

Lemma 3.8

Proof

A nonzero map \(f:M \rightarrow M'\) between 0-generated A-modules with \(\dim _0 M = \dim _0 M'=1\) is surjective, because a proper cokernel would contradict the 0-generated condition. Therefore if \({{\mathrm{Hom}}}_{A}(j_!(N),M) \ne 0\), Lemma 3.6 \((\mathrm {i})\) implies that there is a surjective map \(j_!(N) \twoheadrightarrow M\).

Lemma 4.1

For a finite subgroup \(G\subset {{\mathrm{GL}}}(2,\mathbb {k})\) without pseudo-reflections,  the endomorphism algebra of T is isomorphic to the skew group algebra \(\mathbb {k}[x,y] \rtimes G\).

Proof

Proposition 4.2

For a finite subgroup \(G\subset {{\mathrm{GL}}}(2,\mathbb {k})\) without pseudo-reflections,  the minimal resolution Y of \({\mathbb {A}}^2_\mathbb {k}/G\) is isomorphic to the multigraded linear series \({\mathcal {M}}(E),\) that is, to the fine moduli space of 0-generated A-modules of dimension vector \(\varvec{\mathrm {v}}_{{{\mathrm{Sp}}}}=(\dim (\rho ))_{\rho \in {{\mathrm{Sp}}}(G) }\).

Proof

Since \(Y\cong {\mathcal {M}}(T_{{{\mathrm{Irr}}}})\) and since E is the summand of \(T_{{{\mathrm{Irr}}}}\) corresponding to the cornering subset \({{\mathrm{Sp}}}(G)\subseteq {{\mathrm{Irr}}}(G)\), we need only prove that \(g_{{{\mathrm{Sp}}}}:{\mathcal {M}}(T_{{{\mathrm{Irr}}}}) \rightarrow {\mathcal {M}}(T_{{{\mathrm{Sp}}}})\) is an isomorphism. The line bundle \(\bigotimes _{\rho \in {{\mathrm{Sp}}}(G)} \det (T_\rho )\) has positive degree on each exceptional curve in Y, so it’s ample. Applying Theorem 2.6 and specifically Remark 2.8 shows that \(g_{{{\mathrm{Sp}}}}\) is a closed immersion. It remains to prove that \(g_{{{\mathrm{Sp}}}}\) is surjective.

Remark 4.3

Proposition 4.2 also follows from Karmazyn [23, Corollary 5.4.5], because the reconstruction bundle is a tilting bundle. When \(G\subset {{\mathrm{GL}}}(2,\mathbb {k})\) is cyclic, an explicit construction of the isomorphism \(Y\cong {\mathcal {M}}(A,\varvec{\mathrm {v}}_{{{\mathrm{Sp}}}},\theta )\) for \(\theta \in \Theta _{\varvec{\mathrm {v}}_{{{\mathrm{Sp}}}}}^+\) was first given by [16, 37].

Theorem 4.4

Let \(G\subset {{\mathrm{GL}}}(2,\mathbb {k}),\) be a finite subgroup without pseudo-reflections,  and let \(Y^\prime \) be any partial resolution of \({\mathbb {A}}^2_\mathbb {k}/G\) such that the minimal resolution \(Y\rightarrow {\mathbb {A}}^2_\mathbb {k}/G\) factors via \(Y^\prime \). There is a cornering set \(C \subseteq {{\mathrm{Sp}}}(G)\) containing \(\rho _0\) such that \(Y^\prime \) is isomorphic to \({\mathcal {M}}(E_{C})\).

Proof

The partial resolution \(Y^\prime \) is obtained by contracting a set of components of the exceptional divisor in Y. Remove from \({{\mathrm{Sp}}}(G)\) those \(\rho \) such that \(\det (T_\rho )\) is dual to one of the curves being contracted, leaving a subset \(C\subset {{\mathrm{Sp}}}(G)\) containing \(\rho _0\) such that the contraction \(Y\rightarrow Y^\prime \) contracts precisely the same curves as the morphism \(g_{C}\) from (4.3). The result follows once we prove that \(g_{C}\) is surjective.

Remark 4.5

An analogous result in the complete local setting can be deduced by combining Karmazyn [23, Corollary 5.2.5] with Iyama–Kalck–Wemyss–Yang [26, Theorem 4.6].

Example 4.6

To illustrate Proposition 4.2 and Theorem 4.4, consider the subgroup

$$\begin{aligned} G:=\left\langle \left( \begin{array}{c@{\quad }c} \varepsilon _4 &{} 0 \\ 0 &{}\varepsilon _4^{-1} \end{array} \right) , \left( \begin{array}{c@{\quad }c} 0 &{}\varepsilon _4 \\ \varepsilon _4 &{} 0 \end{array} \right) , \left( \begin{array}{c@{\quad }c} 0 &{}\varepsilon _6 \\ \varepsilon _6 &{} 0 \end{array}\right) \right\rangle \subset {{\mathrm{GL}}}(2,\mathbb {k}), \end{aligned}$$

where \(\varepsilon _r\) is a primitive rth root of unity. This group G is the direct product of the quaternion group of order 8 with a cyclic group of order 3; it has 24 elements and 15 irreducible representations, 3 of dimension two and 12 of dimension one. Below we draw: (a) the McKay quiver (taken with the mesh relations) where we mark the special representations labelled \(0,1,\ldots , 4\); and (b) the Special McKay quiver with the given relations which provides a presentation for the reconstruction algebra A.

Open image in new window

(The mesh relations kill the paths between vertices 0 and 4 that don’t factor through 1, 2, or 3, so there are no arrows in the Special McKay quiver between 0 and 4.) In this case, the exceptional divisor of the minimal resolution \(Y\rightarrow {\mathbb {A}}^2_\mathbb {k}/G\) consists of three \((-2)\)-curves meeting a central \((-3)\)-curve as shown.

Open image in new window

The minimal resolution Y can be constructed either as the G-Hilbert scheme, whose tautological bundle T has 15 non-isomorphic summands, or as the fine moduli space \({\mathcal {M}}(E)\) of 0-generated modules over the reconstruction algebra, in which case the tautological bundle \(\bigoplus _{\rho \in {{\mathrm{Sp}}}(G)} T_\rho \) has 5 non-isomorphic summands.

To illustrate Theorem 4.4, consider the cornering subset \(C=\{0,1,2,3\}\) of the set \({{\mathrm{Sp}}}(G)=\{0,1,2,3,4\}\) of special representations of G. The algebra \(A_{C}\) obtained from the reconstruction algebra A can be presented using the following quiver with relations:

Open image in new window

The morphism \(g_{C}:Y={\mathcal {M}}(E) \rightarrow Y^\prime ={\mathcal {M}}(E_{C})\) is surjective, and it contracts the central (-3)-curve of the exceptional divisor in the minimal resolution.

Assumption 5.1

Remarks 5.2

Remark 5.3

Theorem 5.4

Proof

Remark 5.5

This proof is adapted from that of Proposition A.3 which in turn is adapted from the connectedness result of Bridgeland–King–Reid [5, Section 8]; our use of the numerical Grothendieck group enables us to bypass [5, Lemma 8.1].

Corollary 5.6

Proof

Remark 5.7

In fact, the proof of Corollary 5.6 \((\mathrm {ii})\) shows that one requires only that the class \([\Psi (S_i)]\in K^{{\text {num}}}_c(Y)\) is a non-negative combination of classes of sheaves for each \(i\in Q_0{\setminus } C\).

Proposition 6.1

Let X be a Gorenstein affine toric threefold. For any \(C\subseteq Q_0\) containing \(\{0\},\) the image of \(g_C\) is the irreducible component \(Y_\theta \) of \({\mathcal {M}}(T_C)\) that’s birational to X.

Proof

Example 6.2

For the cyclic subgroup \(G\subset {{\mathrm{SL}}}(3,\mathbb {C})\) of type \(\frac{1}{3}(1,1,1)\), an NCCR for the G-invariant ring \(R:=\mathbb {k}[x,y,z]^{G}\) can be presented as the McKay quiver with relations:

Open image in new window

The cornering subset \(C=\{0,2\}\) corners away the vertex 1 to produce the algebra \(A_{C}\), which can be presented as the quiver

Open image in new window

with relations inherited from A. Consider the \(A_C\) representation \((N,\lambda )\) of dimension vector (1, 1) defined by

$$\begin{aligned} \begin{bmatrix} \lambda _{x_2}&\lambda _{y_2}&\lambda _{z_2} \end{bmatrix} =\begin{bmatrix} 0&0&0 \end{bmatrix} \quad \text {and}\quad \begin{bmatrix} \lambda _{x_1x_0}&\lambda _{x_1 y_0}&\lambda _{x_1 z_0} \\ \lambda _{y_1x_0}&\lambda _{y_1 y_0}&\lambda _{y_1 z_0} \\ \lambda _{z_1x_0}&\lambda _{z_1 y_0}&\lambda _{z_1 z_0} \\ \end{bmatrix}= \begin{bmatrix} 1&\quad 0&\quad 0 \\ 0&\quad 1&\quad 0 \\ 0&\quad 0&\quad 1 \end{bmatrix}. \end{aligned}$$

Then \(j_!(N)\) is an A representation \((V,\rho )\) with dimension vector (1, 3, 1) defined by

$$\begin{aligned} \begin{bmatrix} \rho _{x_2}&\quad \rho _{y_2}&\quad \rho _{z_2} \end{bmatrix} = \begin{bmatrix} 0&0&0 \end{bmatrix},\quad \begin{bmatrix} \rho _{x_0}&\rho _{y_0}&\rho _{z_0} \end{bmatrix} = \begin{bmatrix} 1&\quad 0&\quad 0 \\ 0&\quad 1&\quad 0 \\ 0&\quad 0&\quad 1 \end{bmatrix},\quad \begin{bmatrix} \rho _{x_1} \\ \rho _{y_1} \\ \rho _{z_1} \end{bmatrix} = \begin{bmatrix} 1&\quad 0&\quad 0 \\ 0&\quad 1&\quad 0 \\ 0&\quad 0&\quad 1 \end{bmatrix}. \end{aligned}$$

This representation does not have \(S_1\) as a submodule, and hence the point in \({\mathcal {M}}(E_{C})\) corresponding to N is not in the image of \(g_{C}\) by Proposition 3.7.

Definition 6.3

A vertex \(i\in Q_0\) is essential if there exists a 0-generated A-module that has a submodule isomorphic to \(S_i\). Let \(E\subset Q_0\) denote the set of essential vertices.

Remark 6.4

The proof of Bocklandt–Craw–Quintero Vélez [2, Proposition 4.7] shows that a vertex \(i\in Q_0\) is essential if and only if the 0-generated GIT chamber in \(\Theta _{\varvec{\mathrm {v}}}\) has a wall contained in the hyperplane \(S_i^\perp :=\{\theta \in \Theta _{\varvec{\mathrm {v}}} \mid \theta (S_i)=0\}\). This observation is the key ingredient in the next result which completes the proof of Proposition 1.5.

Proposition 6.5

Let X be a Gorenstein affine toric threefold. For the set \(C:= \{0\}\cup E,\) the morphism \(g_C:Y\rightarrow {\mathcal {M}}(T_C)\) is an isomorphism onto the irreducible component \(Y_\theta \) in \({\mathcal {M}}(T_C)\).

Proof

Remark 6.6

In the special case where \(G\subset {{\mathrm{SL}}}(3,\mathbb {C})\) is a finite abelian subgroup, Proposition 6.5 recovers the main result of Takahashi [34] by reconstructing the G-Hilbert scheme Y as an irreducible component of the fine moduli space of 0-generated \(A_C={{\mathrm{End}}}_Y(T_C)\)-modules of dimension vector \(\varvec{\mathrm {v}}_C\). One cannot strengthen this conclusion, simply because the moduli space \({\mathcal {M}}(T_C)\) need not be irreducible in this case, as shown in Example 6.2.

Lemma 7.1

A nonzero vertex \(i\in Q_0\) is essential iff the object \(\Psi (S_i)\) in \(D^b(A)\) is a sheaf.

Proof

The proof of Bocklandt–Craw–Quintero Vélez [2, Lemma 4.2] shows that a nonzero vertex \(i\in Q_0\) is essential if and only if there exists a torus-invariant 0-generated A-module that has a submodule isomorphic to \(S_i\). Now apply [2, Theorem 1.1\((\mathrm {i})\), Proposition 1.3]. \(\square \)

Remark 7.2

One can also show (see [2, Theorem 1.4]) that each nonzero inessential (= not essential) vertex \(i\in Q_0\) is such that the class \([\Psi (S_i)]\in K^{{\text {num}}}_c(Y)\) is equal to \(-[\mathcal {F}]\), where \(\mathcal {F}\) is the class of a sheaf. In particular, the statement of Corollary 5.6 \((\mathrm {ii})\) does not hold if we remove inessential vertices from \(Q_0\), so it does not apply in the situation of Proposition 6.5.

Proposition 7.3

Let \(C\subseteq {{\mathrm{Irr}}}(G)\) be a subset obtained by removing at most one irreducible representation that marks each proper,  torus-invariant surface in \(G{\text {-Hilb}}\). If \(\dim _i j_!(N_x) \ne 0\) for all \(i\in Q_0{\setminus } C\) and all \(x \in Y',\) then the universal morphism \(g_C:Y\rightarrow {\mathcal {M}}(T_C)\) is an isomorphism.

Proof

Let \(\theta _C\in \Theta _{\varvec{\mathrm {v}}}\) denote the stability parameter defined as in (6.5) above. Again we claim that the line bundle \(L(\theta _C)\cong \bigotimes _{\rho \in C} T_\rho \) on Y is ample. To see this, it suffices to show that for each torus-invariant curve \(\ell \subset Y\), there exists \(\rho \in C\) such that \(\deg T_\rho \vert _\ell >0\). Reid’s recipe labels the curve \(\ell \) with some \(\rho \in {{\mathrm{Irr}}}(G)\), and we have \(\deg T_\rho \vert _\ell = 1\) by [14, Lemma 7.2]. To see that \(\rho \in C\), [14, Corollary 4.6] states that every irreducible representation marks either a (collection of) curve(s) in Y or a unique proper surface, and since we obtain C from \(Q_0={{\mathrm{Irr}}}(G)\) by removing only representations that mark surfaces, we have \(\rho \in C\) after all.

Theorem 2.6 and Proposition 6.1 imply that \(g_C:Y\rightarrow {\mathcal {M}}(T_C)\) is an isomorphism onto the irreducible component \(Y_\theta \) in \({\mathcal {M}}(T_C)\). Since we obtain C by removing only essential vertices from \({{\mathrm{Irr}}}(G)\), the object \(\Psi (S_\rho )\) is a sheaf for each \(\rho \in Q_0{\setminus } C\) by Lemma 7.1. Since \(\dim _i j_!(N_x) \ne 0\) for all \(i\in Q_0{\setminus } C\) and all \(x \in Y'\), Corollary 5.6 \((\mathrm {ii})\) shows that \(g_C\) is an isomorphism. \(\square \)

Remarks 7.4

Example 7.5

For the cyclic subgroup \(G\subset {{\mathrm{SL}}}(3,\mathbb {C})\) of type \(\frac{1}{6}(1,2,3)\), an NCCR for the G-invariant ring \(R:=\mathbb {k}[x,y,z]^{G}\) can be presented as the McKay quiver with relations:

Open image in new window

The cornering set \(C=\{0,1,2,3,4\}\) removes the surface essential vertex 5, and the right \(A_C\)-module \(e_5 A e_C\) is generated by \((x_4, y_3, z_2)\) and can be presented as the cokernel of the matrix

$$\begin{aligned} P=\begin{bmatrix} 0&\quad -z_1&\quad y_2 \\ z_0&\quad 0&\quad -x_2 \\ -y_0&\quad x_1&\quad 0 \end{bmatrix}. \end{aligned}$$

An \(A_C\) representation \((N,\lambda )\) of dimension vector (1, 1, 1, 1, 1) assigns a linear map \(\lambda _p \in Mat (\mathbb {k}) \cong \mathbb {k}\) to each path \(p \in A_C\) such that \(\dim _5 j_!(N)= \dim {{\mathrm{Coker}}}P_N \) where

$$\begin{aligned} P_N:=\begin{bmatrix} 0&\quad -\lambda _{z_1}&\quad \lambda _{y_2} \\ \lambda _{z_0}&\quad 0&\quad -\lambda _{x_2} \\ -\lambda _{y_0}&\quad \lambda _{x_1}&\quad 0 \end{bmatrix}. \end{aligned}$$

The matrix \(P_N\) has determinant of \(\lambda _{x_1} \lambda _{y_2} \lambda _{z_0} - \lambda _{x_2} \lambda _{y_0} \lambda _{z_1}\), and it can be calculated using the relations on \(A_C\) that

$$\begin{aligned} \begin{aligned} \lambda _{x_0x_1} \det P&= \lambda _{x_0x_1}(\lambda _{x_1} \lambda _{y_2} \lambda _{z_0} - \lambda _{x_2} \lambda _{y_0} \lambda _{z_1}) \\&= \lambda _{x_1} \lambda _{x_0x_1y_2} \lambda _{z_0} - \lambda _{x_0x_1x_2} \lambda _{y_0} \lambda _{z_1} \\&= \lambda _{x_1} \lambda _{y_0x_2x_3} \lambda _{z_0} - \lambda _{x_1x_2} \lambda _{y_0} \lambda _{x_0z_1} \\&= \lambda _{x_1x_2} \lambda _{y_0} \lambda _{z_0x_3} - \lambda _{x_1x_2} \lambda _{y_0} \lambda _{z_0x_3} \\&= \lambda _{x_1} \lambda _{x_2} \lambda _{y_0} \lambda _{z_0x_3} - \lambda _{x_1x_2} \lambda _{y_0} \lambda _{z_0} \lambda _{x_3} \\&=0 \end{aligned} \ \begin{aligned} \lambda _{y_0} \det P&= \lambda _{y_0}(\lambda _{x_1} \lambda _{y_2} \lambda _{z_0} - \lambda _{x_2} \lambda _{y_0} \lambda _{z_1}) \\&= \lambda _{x_1y_2} \lambda _{y_0} \lambda _{z_0} - \lambda _{y_0x_2} \lambda _{y_0} \lambda _{z_1} \\&= \lambda _{y_1x_3} \lambda _{y_0} \lambda _{z_0} - \lambda _{x_0y_1} \lambda _{y_0} \lambda _{z_1} \\&= \lambda _{y_1} \lambda _{y_0} \lambda _{z_0x_3} - \lambda _{x_0y_1} \lambda _{y_0} \lambda _{z_1} \\&= \lambda _{y_1} \lambda _{y_0} \lambda _{x_0}\lambda _{z_1} - \lambda _{x_0} \lambda _{y_1} \lambda _{y_0} \lambda _{z_1} \\&=0. \end{aligned} \end{aligned}$$

When N is 0-generated there is a path p from vertex 0 to vertex 2 such that \(\lambda _p \ne 0\). All paths in A from 0 to 2 either factor through vertex 5 or one of the paths \(x_0x_1\) or \(y_0\). As such \(\det P =0\) and \(\dim _5 j_!(N)>0\). It’s easy to compute using Reid’s recipe that 5 is a surface essential vertex, so the G-Hilbert scheme is isomorphic to \({\mathcal {M}}(T_C)\) in this case by Proposition 7.3.

Remark 7.6

For the previous example, vertex 5 is the only essential vertex. It is natural to ask whether the morphism \(g_C:Y\rightarrow {\mathcal {M}}(T_C)\) from Proposition 6.1 is always an isomorphism when we corner away essential vertices. The next example shows that this is not the case when we corner away a pair of essential vertices that mark the same proper torus-invariant surface according to Reid’s recipe. The simplest example of the G-Hilbert scheme with this phenomenon is for an action of the group \((\mathbb {Z}/3\mathbb {Z})^{\oplus 2}\), because in that case the \(G{\text {-Hilb}}\) contains a del Pezzo surface of degree 6 (see [14, Lemma 3.4]). Here we choose a simpler example defined by a consistent dimer model, so in this case we use Reid’s recipe as described by Tapia Amador [33].

Example 7.7

Let \(\text {dP}_6\) denote the del Pezzo surface of degree six, let Y denote the total space of the canonical bundle on \(\text {dP}_6\), and write \(\pi :Y\rightarrow \text {dP}_6\) for the contraction of the zero-section. There is a full strong exceptional collection of globally generated line bundles \((L_0,\ldots , L_5)\) on \(\text {dP}_6\), and the bundle \(\bigoplus _{0\le i\le 5} \pi ^*(L_i)\) is a tilting bundle on Y whose endomorphism algebra can be presented by the following quiver with relations

Open image in new window

The cornering set \(C:=\{ 0,1,2,3,4 \}\) removes the surface essential vertex 5. An \(A_C\) representation \((N,\lambda )\) of dimension vector (1, 1, 1, 1, 1) assigns a linear maps \(\lambda _p \in \mathbb {k}\) to paths in \(A_C\). The right \(A_C\)-module \(e_5Ae_C\) is generated by \((x_5,y_5,z_5)\) and the vector space \(e_5 j_!(N)\) is given by the cokernel of the matrix

$$\begin{aligned} P_N=\begin{bmatrix} 0&\quad -\lambda _{a_1}&\quad \lambda _{b_1}\\ \lambda _{b_2}&\quad 0&\quad -\lambda _{a_2} \\ -\lambda _{a_3}&\quad \lambda _{b_3}&\quad 0 \end{bmatrix} \end{aligned}$$

which has determinant \(\lambda _{b_1} \lambda _{ b_2} \lambda _{b_3}-\lambda _{a_1} \lambda _{ a_2} \lambda _{ a_3}\). Using the relations on \(A_C\) we calculate

$$\begin{aligned} \lambda _{x_4} \det P&= \lambda _{x_4}(\lambda _{b_1} \lambda _{ b_2} \lambda _{b_3}-\lambda _{a_1} \lambda _{ a_2} \lambda _{ a_3}) \\&= (\lambda _{b_1x_4} \lambda _{ b_2} \lambda _{b_3}-\lambda _{a_1x_4} \lambda _{ a_2} \lambda _{ a_3}) \\&= (\lambda _{a_3z_4} \lambda _{ b_2} \lambda _{b_3}-\lambda _{b_2y_4} \lambda _{ a_2} \lambda _{ a_3}) \\&= (\lambda _{a_3} \lambda _{ b_2} \lambda _{b_3z_4}-\lambda _{b_2} \lambda _{ a_2y_4} \lambda _{ a_3}) \\&= (\lambda _{a_3} \lambda _{ b_2} \lambda _{a_2y_4}-\lambda _{b_2} \lambda _{ a_2y_4} \lambda _{ a_3}) =0, \end{aligned}$$

and similarly \( \lambda _{y_4} \det P =0\) and \(\lambda _{z_4} \det P =0\). When N is 0-generated there is a path p from vertex 0 to vertex 4 such that \(\lambda _p \ne 0\). Any such path contains one of \(x_4,y_4\) or \(z_4\) so one of \(\lambda _{x_4}, \lambda _{y_4}\), or \(\lambda _{z_4}\) must be non-zero, so \(\det (P)=0\) and hence \(\dim {{\mathrm{Coker}}}P >0\). In particular \(\dim _5 j_!(N) >0\) and by Proposition 7.3 the cornering morphism \(g_C\) is surjective because 5 is a surface essential.

Note in this case that the tautological bundles \(T_i\) on Y satisfy \(T_4\otimes T_5\cong T_1\otimes T_2\otimes T_3\), and removing both \(T_4\) and \(T_5\) leaves too few bundles to generate the Picard group of Y.