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Multigraded linear series and recollement. (English) Zbl 1396.14005

A common way of studying algebraic varieties is to study their embeddings in projective space by linear series of base-point free line bundles. In this article, this technique is extended to multigraded linear series of a collection of globally generated vector bundles on a scheme, unifying several constructions in the study of algebraic varieties. The primary goal is to describe schemes as a moduli space to achieve geometric properties. The article gives several explicit families of examples.
Throughout the article, \(\Bbbk\) denotes an algebraically closed field of characteristic \(0\). Let \(Y\) be a projective scheme. Given a collection \(E_1,\dots,E_n\) of effective vector bundles on \(Y\), let \(E=\bigoplus_{0\leq i\leq n}E_i\) where \(E_0\simeq\mathcal O_Y\). Let \(A=\text{End}_Y(E)\) be the endomorphism algebra of \(E\) and let \(\mathbf{v}=(v_i)\) with \(v_i=\text{rk}(E_i)\) be the dimension vector. The multigraded linear series of \(E\) is the fine moduli space \(\mathcal M(E)\) of \(0\)-generated \(A\)-modules with dimension vector \(\mathbf{v}\). The universal family of \(\mathcal M(E)\) is a vector bundle \(T=\bigoplus_{0\leq i\leq n}T_i\) together with a \(\Bbbk\)-algebra homomorphism \(A\rightarrow\text{End}(T)\) where each \(T_i\) is a tautological vector bundle of rank \(v_i\) and \(T_0\) is the trivial line bundle.
The first main result of the article generalizes the classical morphism \(\phi_{|L|}:Y\rightarrow |L|\) to the linear series of a single base-point free line bundle \(L\) on \(Y\), i.e. the morphism to a Grassmannian defined by a globally generated vector bundle on a projective variety. Verbatim:
Theorem 1.1. If the vector bundles \(E_1,\dots,E_n\) are globally generated, then there is a morphism \(f:Y\rightarrow\mathcal M(E)\) satisfying \(E_i=f^\ast(T_i)\) for \(0\leq i\leq n\) whose image is isomorphic to the image of the morphism \(\phi_{|L|}:Y\rightarrow|L|\) to the linear series of \(L:=\bigotimes_{1\leq i\leq n}\det(E_i)^{\otimes j}\) for some \(j>0.\)
If the line bundle \(\bigotimes_{1\leq i\leq n}\det(E_i)\) is ample, after taking a multiple of a linearisation if necessary, the resulting universal morphism \(f:Y\rightarrow\mathcal M(E)\) is a closed immersion. The next question is then if \(f\) is surjective. If that is the case, then \(f\) represents \(Y\) as the fine moduli space \(\mathcal M(E)\). It turns out that even when \(Y\) is isomorphic to \(\mathcal M(E),\) more insight can be gained by deleting some summands of \(E.\) When \(0\in C\subseteq\{0,1,\dots,n\},\) the subbundle \(E_C=\sum_{i\in C}E_i\) has the trivial subbundle \(E_0\) as a summand, and Theorem 1.1 proves the universal morphism \(g_C:\mathcal M(E)\rightarrow\mathcal M(E_C)\) between multigraded linear series. This can lead to a more geometric significant moduli space description of \(Y\).
Of such geometric significance is a moduli construction determined by a tilting bundle. A. Bergman and N. J. Proudfoot [Pac. J. Math. 237, No. 2, 201–221 (2008; Zbl 1151.18011)] proved that for a smooth variety \(Y\) with a tilting bundle \(E\), \(f\) is an isomorphism onto a connected component of \(\mathcal M(E)\). A main goal of this article is to establish several cases where \(f\) is an isomorphism onto \(\mathcal M(E)\) itself, implying a description of \(Y\) as a moduli space. Also, the authors manage to give results in situations where \(Y\) is singular.
A second main goal is to apply the theory to the special McKay correspondence. Let \(G\subset\text{GL}(2,\Bbbk)\) be a finite subgroup without pseudo-reflections, let \(\text{Irr}(G)\) be the set of isomorphism classes of irreducible representations of \(G\), and let \(Y\) denote the minimal resolution of \(\mathbb A^2_{\Bbbk}/G\). R. Kidoh [Hokkaido Math. J. 30, No. 1, 91–103 (2001; Zbl 1015.14004)] and A. Ishii [J. Reine Angew. Math. 549, 221–233 (2002; Zbl 1057.14057)] proved that \(Y\) is isomorphic to the fine moduli space of \(G\)-equivariant coherent sheaves of the form \(\mathcal O_Z\), for subschemes \(Z\subset\mathbb A^2_{\Bbbk}\) such that \(\Gamma(\mathcal O_Z)\) is isomorphic to the regular representation of \(G\) (\(G\)-Hilbert scheme). Writing the tautological bundle on the \(G\)-Hilbert space \(T=\underset{\rho\in\text{Irr}(G)}{\sum} T_\rho^{\bigoplus\dim(\rho)}\) and noticing that \(\text{End}_Y(T)\) is isomorphic to the skew group algebra, it follows that the minimal resolution \(Y\cong G-\text{Hilb}\) is isomorphic to the multigraded linear series \(\mathcal M(T)\). When \(G\) is a finite subgroup of \(\text{SL}(2,\Bbbk)\), \(T\) is a tilting bundle on \(Y\) so that \(Y\) is derived equivalent to the category of modules over \(\text{End}_Y(T),\) but the authors prove that this is false in general.
A natural moduli description of \(Y\) is given by the special McKay correspondence of the finite subgroup \(G\subset\text{GL}(2,\Bbbk)\). For the set \(\text{Sp}(G)=\{\rho\in\text{Irr}(G)|H^1(T^\vee_\rho)=0\}\) of special representations, Van den Bergh proved that the reconstruction bundle \(E:=\underset{\rho\in\text{Sp}(G)}{\bigoplus} T_\rho\) is a tilting bundle on \(Y\) so that \(Y\) is derived equivalent to the category of modules over the reconstruction algebra studied by M. Wemyss [Math. Ann. 350, No. 3, 631–659 (2011; Zbl 1233.14012)]. The second main result of the article proves that \(E\) contains enough information to reconstruct \(Y\), and provides a moduli space description that trumps the \(G\)-Hilbert scheme in general. Verbatim:
Theorem 1.2. Let \(G\subset\text{GL}(2,\Bbbk)\) be a finite subgroup without pseudo-reflections. Then: (i) the minimal resolution \(Y\) of \(\mathbb A^2_{\Bbbk}/G\) 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 \(Y\rightarrow\mathbb A^2_\Bbbk/G\) factors via \(Y^\prime,\) there is a summand \(E_C\subseteq E\) such that \(Y^\prime\) is isomorphic to \(\mathcal M(E_C).\)
The authors correctly remark that the approach in this article is closer in spirit to the geometric construction of the special McKay correspondence for cyclic subgroups of \(\text{GL}(2,\Bbbk)\) given by A. Craw [Q. J. Math. 62, No. 3, 573–591 (2011; Zbl 1231.14010)] in his earlier work.
The main tools for proving the theorems is a homological criterion to decide when the morphism \(g_C\) is surjective. Any subset \(C\subseteq\{0,\dots,n\}\) containing \(0\) determines a subbundle \(E_C\) of \(E\), and the module categories of the algebras \(A=\text{End}_Y(E)\) and \(A_C=\text{End}_Y(E_C)\) are linked by recollement. The authors prove that the morphism \(g_C:\mathcal M(E)\rightarrow\mathcal M(E_C)\) is surjective iff for each \(c\in\mathcal M(E_C)\), the \(A\)-module \(j_!(N_x)\) admits a surjective map onto an \(A\)-module of dimension vector \(\mathbf(v)\).
As a second main ingredient, the authors use the fact that a derived equivalence \(\Psi(-)=E^\vee\otimes_A-:D^b(A)\rightarrow D^b(Y)\) induces an isomorphism between the lattice of dimension vectors for \(A\) and the numerical Grothendieck group for compact support \(K_c^{\text{num}}(Y)\) introduced by Bayer-Craw-Zhang.
The article concludes with examples from NCCRs in dimension three. In very short terms, a resolution \(Y\rightarrow\mathbb A/G\) is given. Examples are given where one can reconstruct \(Y\) using only a proper summand of a tilting bundle \(T\).
The article is very important, contains very nice results with clear proofs, and show important applications of tilting- and moduli theory, and their connection.

MSC:

14A22 Noncommutative algebraic geometry
14E16 McKay correspondence
16G20 Representations of quivers and partially ordered sets
18F30 Grothendieck groups (category-theoretic aspects)
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References:

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