Multigraded linear series and recollement
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Abstract
Given a scheme Y equipped with a collection of globally generated vector bundles \(E_1, \ldots , E_n\), we study the universal morphism from Y to a fine moduli space \({\mathcal {M}}(E)\) of cyclic modules over the endomorphism algebra of \(E:={\mathcal {O}}_Y\oplus E_1\oplus \cdots \oplus E_n\). This generalises the classical morphism to the linear series of a basepoint-free line bundle on a scheme. We describe the image of the morphism and present necessary and sufficient conditions for surjectivity in terms of a recollement of a module category. When the morphism is surjective, this gives a fine moduli space interpretation of the image, and as an application we show that for a small, finite subgroup \(G\subset {{\mathrm{GL}}}(2,\mathbb {k})\), every sub-minimal partial resolution of \({\mathbb {A}}^2_\mathbb {k}/G\) is isomorphic to a fine moduli space \({\mathcal {M}}(E_C)\) where \(E_C\) is a summand of the bundle E defining the reconstruction algebra. We also consider applications to Gorenstein affine threefolds, where Reid’s recipe sheds some light on the classes of algebra from which one can reconstruct a given crepant resolution.
Keywords
Linear series Moduli space of quiver representations Special McKay correspondence Noncommutative crepant resolutionsMathematics Subject Classification
Primary 14A22 Secondary 14E16 16G20 18F301 Introduction
The study of an algebraic variety in terms of the morphisms to the linear series of basepoint-free line bundles has always been a central tool in algebraic geometry. Here we extend this notion to the multigraded linear series of a collection of globally generated vector bundles on a scheme, thereby unifying several constructions from the literature (see [16, 17] and [15, Section 5]). Our primary goal is to provide new, geometrically significant moduli space descriptions of any given scheme, and we illustrate this in several families of examples.
Multigraded linear series To be more explicit, let Y be a scheme that is projective over an affine scheme of finite type over \(\mathbb {k}\), an algebraically closed field of characteristic zero. Given a collection \(E_1, \ldots , E_n\) of effective vector bundles on Y, define \(E:=\bigoplus _{0\le i\le n} E_i\) where \(E_0\) is the trivial line bundle on Y. Let \(A:={{\mathrm{End}}}_Y(E)\) denote the endomorphism algebra and consider the dimension vector \(\varvec{\mathrm {v}}:=(v_i)\) given by \(v_i:={{\mathrm{rk}}}(E_i)\) for \(0\le i\le n\). We define the multigraded linear series of E to be the fine moduli space \({\mathcal {M}}(E)\) of 0-generated A-modules of dimension vector \(\varvec{\mathrm {v}}\) (see Definition 2.5). The universal family on \({\mathcal {M}}(E)\) is a vector bundle \(T=\bigoplus _{0\le i\le n} T_i\) together with a \(\mathbb {k}\)-algebra homomorphism \(A\rightarrow {{\mathrm{End}}}(T)\), where \(T_i\) is a tautological vector bundle of rank \(v_i\) for \(1\le i\le n\) and \(T_0\) is the trivial line bundle.
Our first main result (see Theorem 2.6) generalises the classical morphism \(\varphi _{\vert L\vert }:Y\rightarrow \vert L\vert \) to the linear series of a single basepoint-free line bundle L on Y, or the morphism to a Grassmannian defined by a globally generated vector bundle on a projective variety:
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 \(j>0\).
A statement similar to Theorem 1.1 holds if we replace \({\mathcal {M}}(E)\) by a product of Grassmannians over \(\Gamma ({\mathcal {O}}_Y)\). However, the dimension of this product is higher than that of \({\mathcal {M}}(E)\) in general, and f is almost never an isomorphism, i.e., f does not provide a moduli space description of Y.
A moduli construction determined by a tilting bundle is of clear geometric significance. For a smooth variety Y admitting a tilting bundle E, Bergman–Proudfoot [8, Theorem 2.4] showed that f is an isomorphism onto a connected component of \({\mathcal {M}}(E)\). The goal of this paper is to establish several situations in which f is an isomorphism onto \({\mathcal {M}}(E)\) itself, thereby giving a moduli space description of Y. We do not assume that Y is smooth (it is singular in Theorem 1.2 \((\mathrm {ii})\)), and while E may be a tilting bundle, we do not demand this much; after all, one does not require every indecomposable summand of Beilinson’s tilting bundle in order to reconstruct \(\mathbb {P}^n\).
Theorem 1.2
- (i)
the minimal resolution Y of \({\mathbb {A}}^2_\mathbb {k}/G\) is isomorphic to the multigraded linear series \({\mathcal {M}}(E)\) of the reconstruction bundle; and
- (ii)
for any partial resolution \(Y^\prime \) such that the minimal resolution \(Y\rightarrow {\mathbb {A}}^2_\mathbb {k}/G\) factors via \(Y^\prime ,\) there is a summand \(E_C\subseteq E\) such that \(Y^\prime \) is isomorphic to \({\mathcal {M}}(E_{C})\).
In other words, for a finite subgroup \(G\subset {{\mathrm{GL}}}(2,\mathbb {k})\), the minimal resolution of \({\mathbb {A}}^2_\mathbb {k}/G\) can be obtained directly from the special representations. The statement of part \((\mathrm {i})\) is due originally to Karmazyn [23, Corollary 5.4.5], while an analogue of part \((\mathrm {ii})\) in the complete local setting can be deduced by combining Iyama–Kalck–Wemyss–Yang [26, Theorem 4.6] with [23, Corollary 5.2.5]. Note however that our approach is completely different in each case, and is closer in spirit to the geometric construction of the Special McKay correspondence for cyclic subgroups of \({{\mathrm{GL}}}(2,\mathbb {k})\) given by Craw [16].
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.
Examples from NCCRs in dimension three The morphisms \(Y\rightarrow {\mathbb {A}}^2/G\) from Theorem 1.2 all have fibres of dimension at most one, but our methods apply without this assumption. To illustrate this, we also study crepant resolutions of Gorenstein affine threefolds. For any such singularity X, Van den Bergh’s construction [35] of an NCCR (satisfying Assumption 5.1) produces a crepant resolution of X as a fine moduli space of stable representations for an algebra A (see Proposition A.3). The choice of 0-generated stability condition chooses a particular crepant resolution Y and a globally generated bundle T on Y. In this case, Y is isomorphic to the multigraded linear series \({\mathcal {M}}(T)\); since T is a tilting bundle, this moduli construction is certainly geometrically significant.
Nevertheless, motivated by work of Takahashi [34], we ask whether one can reconstruct Y using only a proper summand of T (in general, none of the indecomposable summands of T is ample). To state the result, we say that a vertex \(i\in Q_0=\{0,1,\ldots , n\}\) is essential if there is a 0-generated A-module of dimension vector \(\varvec{\mathrm {v}}\) that contains the vertex simple A-module \(S_i\) in its socle. The following result combines Propositions 6.1 and 6.5:
Proposition 1.5
- (i)
for any subset \(C\subseteq Q_0\) containing 0, the image of \(g_C:{\mathcal {M}}(T)\rightarrow {\mathcal {M}}(T_C)\) is an irreducible component of \({\mathcal {M}}(T_C);\) and
- (ii)
if C is the union of \(\{0\}\) with the set of essential vertices, then \(g_C\) is an isomorphism onto its image.
Here, part \((\mathrm {ii})\) generalises the result of Takahashi [34] beyond the case where X is an abelian quotient. Example 6.2 shows that Proposition 1.5 is optimal: in general \({\mathcal {M}}(T_C)\) is reducible.
We conclude with several examples that are orthogonal in spirit to Proposition 1.5 \((\mathrm {ii})\), with a view to strengthening the statement to an isomorphism \({\mathcal {M}}(T)\cong {\mathcal {M}}(T_C)\). Rather than keep the summands of T corresponding to essential vertices, instead we use Reid’s recipe as a guide to help us choose which essential vertices to remove. For an essential vertex \(i\in Q_0\), derived Reid’s recipe [2, 11, 12, 28] proves that the image under the derived equivalence \(\Psi \) of the vertex simple A-module \(S_i\) is a sheaf. This gives enough information to compute the class of \(\Psi (j_!(N_x))\) in \(K^{{\text {num}}}_c(Y)\), and under a dimension condition (see Corollary 5.6), we can apply Theorem 1.4 to deduce that \(g_C:{\mathcal {M}}(T)\rightarrow {\mathcal {M}}(T_C)\) is an isomorphism. We showcase this construction in Examples 7.5 and 7.7, and in the latter example we also show that the dimension condition can fail if we remove two indecomposable summands of T corresponding to essential vertices that mark the same surface by Reid’s recipe. Put more geometrically, surjectivity can fail if the summands of \(T_C\) do not generate the Picard group of \({\mathcal {M}}(T)\).
Notation Let \(\mathbb {k}\) be an algebraically closed field of characteristic zero. For any quasiprojective \(\mathbb {k}\)-scheme Y and \(\mathbb {k}\)-algebra A, we write \(D^b(Y)\) and \(D^b(A)\) for the bounded derived categories of coherent sheaves on Y and finitely generated left A-modules respectively. A vector bundle is a locally-free sheaf of finite rank.
2 Multigraded linear series
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
- (i)
The pullback via the universal morphism \(f:Y\rightarrow {\mathcal {M}}(A,\varvec{\mathrm {v}},\theta )\) of the ample bundle \(L_\theta \) induced by the GIT construction is the line bundle \(\bigotimes _{0\le i\le n} \det (E_i)^{\otimes \theta _i}\) on Y; and
- (ii)
if \(L_\theta \) is very ample (which holds after replacing \(\theta \) by a positive multiple if necessary), then the image of \(f:Y\rightarrow {\mathcal {M}}(A,\varvec{\mathrm {v}},\theta )\) is isomorphic to the image of the morphism from Y to the linear series of the globally generated line bundle \(\bigotimes _{0\le i\le n} \det (E_i)^{\otimes \theta _i}\).
Proof
The problem with Lemma 2.2 is that it is a difficult problem in general to find a suitable parameter \(\theta \in \Theta _{\varvec{\mathrm {v}}}\) for which a given vector bundle E defines a flat family of \(\theta \)-stable A-modules.
Here we highlight a special situation where this problem has a simple solution. It’s easy to see that any stability parameter \(\theta =(\theta _i)\in \Theta _{\varvec{\mathrm {v}}}\) satisfying \(\theta _i>0\) for all \(i\ne 0\) is generic, so there is a GIT chamber \(\Theta _{\varvec{\mathrm {v}}}^+\subset \Theta _{\varvec{\mathrm {v}}}\) containing all such stability parameters. Given an A-module M that is \(\theta \)-stable for \(\theta \in \Theta ^+_{\varvec{\mathrm {v}}}\), it follows directly from the definition that there exists a surjective A-module homomorphism \(Ae_0\rightarrow M\). More generally, we say that an A-module M is 0-generated if there exists a surjective A-module homomorphism \(Ae_0\rightarrow M\). It is sometimes advantageous to use this latter notion because it is well-defined without having to make explicit reference to a dimension vector \(\varvec{\mathrm {v}}\).
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
Corollary 2.4 implies that each direct summand of the tautological bundle \(T=\bigoplus _{0\le i\le n} T_i\) on \({\mathcal {M}}(E)\) is globally generated. Moreover, \(L_\theta = \bigotimes _{1\le i\le n} \det (T_i)^{\otimes j}\) is very ample. To justify the terminology ‘multigraded linear series’, we present the following result (compare Example 2.7).
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
- (1)
The construction of [17] is phrased in terms of a quiver with relations that gives a presentation \(A\cong \mathbb {k}Q/I\), leading to an explicit GIT description of the image of f in a quiver flag variety. However, we now assume only that Y is projective over an affine, and in this generality no such natural presentation exists, leading us to place greater emphasis on A rather than on a quiver. While we sacrifice an explicit description of the image of f, Theorem 2.6 nevertheless determines the image of f up to isomorphism.
- (2)
The observation in Theorem 2.6 that the image of f is determined by \(\bigotimes _{1\le i\le n} \det (E_i)^{\otimes j}\) for some \(j> 0\) (see also Remark 2.8) renders redundant the assumption from [17, Corollary 4.10] (and hence from [15, Proposition 5.5] and Prabhu-Naik [31, Proposition 5.5]) that a map obtained by multiplication of global sections is surjective.
3 The cornering category and recollement
In this section we use Theorem 2.6 repeatedly to produce a compatible family of morphisms between different multigraded linear series. We then introduce a homological criterion that is sufficient to guarantee that any of these morphisms is surjective.
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. \(\square \)
Our next result encodes the fact that the morphisms (3.1) are compatible as we vary the choice of the subset C. To state the result, regard the poset of subsets of \(\{0,1,\ldots ,n\}\) that contain \(\{0\}\) as a category \(\mathscr {C}\) in which the morphisms are the set-theoretic inclusion maps between subsets.
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
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. \(\square \)
Of particular interest to us are the following morphisms.
Definition 3.4
Theorem 2.6 ensures that we understand the image of each cornering morphism. The next example illustrates that while these morphisms may be surjective, they need not be.
Example 3.5
Lemma 3.6
- (i)
If N is a 0-generated \(A_C\)-module, then \(j_!N\) is a 0-generated A-module.
- (ii)
The A-module \(j_! N\) is finite-dimensional and satisfies \(\dim _i j_!N = \dim _i N\) for \(i \in C\).
Proof
Note that each closed point \(x\in {\mathcal {M}}(E_C)\) determines a 0-generated \(A_C\)-module \(N_x\) of dimension vector \(\varvec{\mathrm {v}}_C\).
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 \(g_C(y) =x\). \(\square \)
The following result provides a homological criterion to check whether Proposition 3.7 applies.
Lemma 3.8
- (i)
if \({{\mathrm{Hom}}}_{A}(j_!(N),M) \ne 0,\) then there exists a surjective map \(j_!(N) \twoheadrightarrow M;\) and
- (ii)
if \({{\mathrm{Hom}}}_{A}(M,j_!(N)) \ne 0,\) then there exists an isomorphism \(M \cong j_!(N)\).
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 \(j_!(N) \twoheadrightarrow M\).
4 Cornering the reconstruction algebra
Let \(G\subset {{\mathrm{GL}}}(2,\mathbb {k})\) be a finite subgroup that acts without pseudo-reflections. We now apply the results of the previous section to obtain a fine moduli space description of any partial resolution \(Y^\prime \) of a quotient surface singularity \({\mathbb {A}}^2_\mathbb {k}/G\) such that the minimal resolution \(Y\rightarrow {\mathbb {A}}^2_\mathbb {k}/G\) factors through \(Y^\prime \). In fact we prove that every such \(Y^\prime \) is the multigraded linear series for a summand of the tautological bundle on the G-Hilbert scheme.
Let \({{\mathrm{Irr}}}(G)\) denote the set of isomorphism classes of irreducible representations of G. The G-Hilbert scheme is the fine moduli space of G-equivariant coherent sheaves of the form \({\mathcal {O}}_Z\) for \(Z\subset {\mathbb {A}}^2_\mathbb {k}\) such that \(\Gamma ({\mathcal {O}}_Z)\) is isomorphic to the regular representation of G. The category of G-equivariant coherent sheaves is equivalent to the category of finitely generated modules over the skew group algebra \(\mathbb {k}[x,y]\rtimes G\), so the G-Hilbert scheme is isomorphic to the fine moduli space \({\mathcal {M}}(\mathbb {k}[x,y]\rtimes G,\varvec{\mathrm {v}},\theta )\), where \(\varvec{\mathrm {v}}=(\dim (\rho )^{\oplus \dim (\rho )})_{\rho \in {{\mathrm{Irr}}}(G)}\) and \(\theta \in \Theta _{\varvec{\mathrm {v}}}^+\) is a 0-generated stability condition; here the trivial representation \(\rho _0\) is the zero vertex.
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
5 Cornering noncommutative crepant resolutions
In this section we recall Van den Bergh’s notion of an NCCR [35] and show that the moduli spaces determined by 0-generated stability parameters are multigraded linear series. We establish a set of sufficient conditions for the cornering morphisms to be surjective, and we demonstrate that these conditions hold in a range of situations. Throughout this section, we assume that R is a normal, Gorenstein \(\mathbb {k}\)-algebra of Krull dimension at most three.
Recall that a \(\mathbb {k}\)-algebra A is a noncommutative crepant resolution (NCCR) of R if there exists a reflexive R-module M such that \(A:={{\mathrm{End}}}_R(M)\) is Cohen–Macaulay as an R-module and is of finite global dimension. Choose a decomposition \(M \cong \bigoplus _i M_i\) into finitely many indecomposable, reflexive R-modules; in general this decomposition is non-unique. Given one such decomposition \(A= {{\mathrm{End}}}_R(\bigoplus _{i}M_i)\), we obtain a presentation as a quotient \(A\cong \mathbb {k}Q/I\) exactly as for the geometric setting described following Eq. (2.1). We impose the following additional standing assumption on \(A={{\mathrm{End}}}_R(M)\).
Assumption 5.1
- (i)
All indecomposable projective A-modules occur as summands of A, and the presentation \(A=\mathbb {k}Q/I\) corresponds to a unique decomposition of \(M= \bigoplus _{i \in Q_0} M_i\) into non-isomorphic, indecomposable reflexive modules; and
- (ii)
the ideal \(I\subset \mathbb {k}Q\) is generated by linear combinations of paths of length at least one.
Remarks 5.2
- (1)
If the module category of R has the Krull–Schmidt property, then any NCCR \(A'\) is Morita equivalent to an NCCR A satisfying Assumption 5.1 \((\mathrm {i})\).
- (2)
Assumption 5.1 \((\mathrm {ii})\) ensures that each vertex \(i\in Q_0\) determines a vertex simple A-module \(S_i=\mathbb {k}e_i\), and that distinct summands of M are non-isomorphic (as \(M_i\cong M_j\) would force the relations \(aa^\prime -e_i, a^\prime a-e_j\in I\) for some \(a, a^\prime \in Q_1\)).
Define the dimension vector \(\varvec{\mathrm {v}}= (v_i)\in \mathbb {Z}^{Q_0}\) by setting \(v_i={{\mathrm{rk}}}_R(M_i)\) for \(0\le i\le n\). After replacing A by a Morita equivalent algebra if necessary, Van den Bergh [35, Section 6.3] notes that we may assume that \(\varvec{\mathrm {v}}\) is indivisible. For any generic stability parameter \(\theta \in \Theta _{\varvec{\mathrm {v}}}\), the tautological bundle \(T=\bigoplus _{0\le i\le n}T_i\) on the moduli space \({\mathcal {M}}(A,\varvec{\mathrm {v}},\theta )\) is a left A-module, so \(T^\vee \) is a right A-module. When A satisfies Assumption 5.1, [35, Remark 6.6] observes that the approach of Bridgeland–King–Reid [5] implies that \({\mathcal {M}}(A,\varvec{\mathrm {v}},\theta )\) is connected; we provide a slightly simplified proof of this observation in Proposition A.3.
Remark 5.3
- (1)
if more than one summand \(M_i\) of M has rank one, then by relabelling i to be the zero vertex and repeating the construction, the tautological bundle \(T^\prime \) on the resulting moduli space determines a projective crepant resolution \(f^\prime :{\mathcal {M}}(T^\prime )\rightarrow \mathrm{Spec}\,R\) that need not be isomorphic to that from (5.4), e.g. see the suspended pinch point example considered in [7, Example B.8.7] where the choice of zero vertex as the vertex currently labelled 1 or 2 yields two crepant resolutions that are not isomorphic as varieties;
- (2)
for a finite subgroup \(G\subset {{\mathrm{SO}}}(3)\), the skew group algebra \(A=\mathbb {k}[x,y,z]\rtimes G\) is an NCCR of \(R:=\mathbb {k}[x,y,z]^G\). Nolla de Celis–Sekiya [30] show that every projective crepant resolution of \(\mathrm{Spec}\,R\) is of the form \({\mathcal {M}}(A^\prime , \varvec{\mathrm {v}}^\prime , \theta ^\prime )\), where \(A^\prime \) is an algebra obtained from A by a sequence of mutations at vertices (none of which is the zero vertex), where \(\varvec{\mathrm {v}}\) is a dimension vector and where \(\theta ^\prime \) is a 0-generated stability condition. In particular, since the endomorphism algebra of the tautological bundle on \({\mathcal {M}}(A^\prime , \varvec{\mathrm {v}}^\prime , \theta ^\prime )\) is isomorphic to \(A^\prime \), the multigraded linear series of the tautological bundle is isomorphic to \({\mathcal {M}}(A^\prime , \varvec{\mathrm {v}}^\prime , \theta ^\prime )\) and every projective crepant resolution of \(\mathrm{Spec}\,R\) can be constructed as such a multigraded linear series; and
- (3)
if R is a complete local Gorenstein algebra over the field \(\mathbb {C}\) such that \(\mathrm{Spec}\,R\) admits a projective crepant resolution whose fibres have dimension at most one, then as in (2) above, work of Wemyss [39] implies that every projective crepant resolution of \(\mathrm{Spec}\,R\) is isomorphic to a multigraded linear series of a tautological bundle.
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].
This is particularly applicable when the vertex simple A-modules \(S_i\) for \(i \in Q_0 \backslash C\) are well understood and when the dimension of \(j_!(N_x)\) can be calculated.
Corollary 5.6
- (i)
\(\dim _i j_!(N_x) = v_i\) for all \(i \in Q_0{\setminus } C\) and all \(x \in Y';\) or
- (ii)
\(\Psi (S_i)\) is a sheaf for all \(i \in Q_0{\setminus } C\) and \(\dim _i j_!(N_x) \ge v_i\) for all \(i\in Q_0{\setminus } C\) and all \(x \in Y'\).
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 \(i\in Q_0{\setminus } C\).
6 The toric case in dimension three
We now specialise to the case where R is a Gorenstein, semigroup algebra of dimension three, so \(X=\mathrm{Spec}\,R\) is a Gorenstein toric threefold. Ishii–Ueda [20] and Broomhead [9] show that R admits a noncommutative crepant resolution \(A={{\mathrm{End}}}_R(\bigoplus _{i\in Q_0} M_i)\) obtained as the Jacobian algebra \(\mathbb {k}Q/I\) of a quiver with potential arising from a consistent dimer model on a real two-torus. Toric algebras of this form necessarily satisfy Assumption 5.1; in fact, the conclusions of Lemma A.1 were noted first by Ishii–Ueda [22, Proposition 8.3] in this context.
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
The next example shows that \({\mathcal {M}}(T_C)\) need not be irreducible, or equivalently, the morphism \(g_C\) need not be surjective in this context.
Example 6.2
In order to complete the statement and proof of Proposition 1.5, we require a generalisation to our context of a notion introduced by Takahashi [34]:
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.
7 On Reid’s recipe and surface essentials
We continue to work under the assumptions of the previous section, where X is a Gorenstein affine toric threefold and \(Y\cong {\mathcal {M}}(T)\) is a crepant resolution. Our goal is to present a general framework and some explicit examples that are orthogonal in spirit to Proposition 6.5, with a view to obtaining an isomorphism \({\mathcal {M}}(T)\cong {\mathcal {M}}(T_C)\). The terminology ‘essentials’ might suggest that keeping such vertices is crucial if \(g_C\) is to be an isomorphism, but this is not the case; indeed, here we choose which essential vertices to remove from the set \(Q_0\).
Our motivation comes from comparing Corollary 5.6 \((\mathrm {ii})\) with the derived category statement of Reid’s recipe [2, 11, 12, 28]. The key observation is the following (see Remark 6.4 for another equivalent condition).
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.
We now restrict ourselves to the case when X is isomorphic to the quotient singularity \({\mathbb {A}}^3_\mathbb {k}/G\), where \(G\subset {{\mathrm{SL}}}(3,\mathbb {C})\) is a finite abelian subgroup, and where Y is the G-Hilbert scheme (see Remark 7.4 \((\mathrm {i})\)). Let \({{\mathrm{Irr}}}(G)\) denote the set of isomorphism classes of irreducible representation of G; this is the vertex set of the McKay quiver. Reid’s recipe [14, 32] marks every proper, torus-invariant curve and surface in the G-Hilbert scheme Y with an irreducible representation of G; those surfaces that are isomorphic to the del Pezzo surface of degree six are marked with two irreducible representations.
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
- (1)
In Proposition 7.3 we assumed that \(X={\mathbb {A}}^3_\mathbb {k}/G\) (and hence \(Y\cong G{\text {-Hilb}}\)) because [14, Corollary 4.6] is known to hold only for the G-Hilbert scheme at present.
- (2)
The condition in Proposition 7.3 that we remove from \({{\mathrm{Irr}}}(G)\) at most one irreducible representation that marks each surface implies that the indecomposable summands of \(T_C\) generate the Picard group of Y; see [14, Theorem 6.1]. Example 7.7 illustrates that the statement of Proposition 7.3 can fail without this condition.
Example 7.5
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
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.
Footnotes
Notes
Acknowledgements
A. Craw thanks Stefan Schröer for a stimulating discussion. We thank the anonymous referees for pointing out an error in an earlier version of this paper and for helpful comments. A. Craw and J. Karmazyn were supported by EPSRC Grants EP/J019410/1 and EP/M017516/1 respectively, and Y. Ito was supported by JSPS Grant-in-Aid (C) No. 23540045.
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