Non-commutative crepant resolutions, Ulrich modules and generalizations of the McKay correspondence
Title, abstract and references

13th June 10:00-10:10

 Yukari Ito (Nagoya)  Introduction
 Abstract  I will introduce the motivation of this conference and relationship between noncommutative crepant resolution, Ulrich modules and generalization of the McKay corespondence.
 References  Mindmap


 Akira Ishii (Hiroshima)  Introduction to the McKay correspondence and Artin-Verdier theory
 Abstract For a finite subgroup G of SL(2, C), the quotient singularity C^2/G has a resolution whose dual graph is a Dynkin graph of type ADE. In the late 70’s, McKay observed that one can obtain the affine Dynkin graph of the same type by using the representation theory of G. Here, the vertices correspond to irreducible representations of G and thus the existence of a natural correspondence between exceptional curves and representations was expected. In 1984, Artin and Verdier gave a conceptual proof of the existence of such a correspondence. Moreover, as an appendix to the Artin-Verdier paper, Esnault and Knorrer gave an explanation for the correspondence of edges. I will talk about these old results and explain how they can be used to describe the correspondence between derived categories. 
 References [1] J. McKay, Graphs, singularities and finite groups, Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R. I., 1980, pp. 183-186.
[2] Artin, M. and Verdier, J.-L., Reflexive modules over rational double points, Math. Ann. 270 (1985), 79–82
[3] Esnault, H. and Knörrer, H., Reflexive modules over rational double points, Math. Ann. 272 (1985), 545–548.
[4] Esnault, H., Reflexive modules on quotient surface singularities, J. reine angew. Math. 362 (1985), 63–71.
[5] Wunram, J., Reflexive modules on quotient surface singularities, Math. Ann. 279 (1988), 583–598.
[6] Ishii, A., On the McKay correspondence for a finite small subgroup of GL(2,ℂ). J. Reine Angew. Math. 549 (2002), 221–233.
[7] M. Van den Bergh, Three-dimensional flops and noncommutative rings, Duke Math. J. 122 (2004), no. 3, 423–455.


 Yujiro Kawamata (Tokyo)  Derived McKay correspondence for finite abelian groups and derived toric MMP
 Abstract  We review the connection between the toric MMP (minimal model program) and the corresponding sequence of SOD (semi-orthogonal decompositions) of the derived categories, yielding the derived McKay correspondence for finite abelian groups.
 References [1] D-equivalence and K-equivalence, math.AG/0205287, J. Diff. Geom. 61 (2002), 147--171.
[2] Derived categories of toric varieties, Michigan Math. J. 54 (2006), 517--535.
[3] Derived categories of toric varieties II, Michigan Math. J. 62-2 (2013), 353--363.
[4] Derived categories of toric varieties III, European Journal of Mathematics, 2 (2016), 196--207.


 Kazushi Ueda (Tokyo)  Dimer models
 Abstract  We give an introduction to the theory of dimer models, with emphasis on non-commutative crepant resolutions
and derived categories of coherent sheaves.
 References [1] Nathan Broomhead, Dimer models and Calabi-Yau algebras, Memoirs of the American Mathematical Society, Volume 215, Number 1011, viii+86 pages, 2012, arXiv:0901.4662
[2] Raf Bocklandt, A Dimer ABC, arXiv:1510.04242


 Alastair Craw (Bath)  The McKay correspondence by variation of GIT quotient
 Abstract  For a finite subgroup G of SL(3,C), Bridgeland, King and Reid's proof of the McKay correspondence appeared to single out Nakamura's Hilbert scheme of G-clusters as a preferred minimal resolution of the singularity C^3/G. I'll talk about an old paper with Akira Ishii where we demonstrated, at least for G abelian, that every projective minimal resolution can be constructed as a fine moduli space of certain stable modules over the skew group algebra of G (called G-constellations). Recently, Ishii and Ueda extended these results from C^3/G to any Gorenstein affine toric 3-fold. 
[1] Alastair Craw and Akira Ishii, Flops of G-Hilb and equivalences of derived categories by variation of GIT quotient, Duke Math. J. 124 (2004), no.2, 259-307.
[2] Akira Ishii and Kazushi Ueda, Dimer models and crepant resolutions, arXiv:1303.4028.


 Hokuto Uehara (TMU)  A trichotomy for the autoequivalence groups of derived categories on surfaces
 Abstract  We study the group of autoequivalences of the derived categories on smooth projective surfaces, and show a trichotomy of types of the groups, that is, of K3 type, of elliptic surface type and of general type. We also pose a conjecture on the description of each type of the groups, and prove it in some special cases.
 References  (in preparation)

14th June 9:30-10:15

 Ryo Takahashi (Nagoya)  Introduction to Cohen-Macaulay representation theory
 Abstract  The main subject of Cohen-Macaulay representation theory is to study the structure of (maximal) Cohen-Macaulay modules over a given Cohen-Macaulay ring. In this talk, I will roughly explain the so-called "classical period" of Cohen-Macaulay representation theory, which concerns Cohen-Macaulay local commutative rings and has been established by 1990. If time permits, I would like to mention a celebrated result of Auslander on isolated singularities and speak about a recent progress.
 References [1] M. Auslander, Isolated singularities and existence of almost split sequences, Representation theory, II (Ottawa, Ont., 1984), 194--242, Lecture Notes in Math., 1178, Springer, Berlin, 1986.
[2] H. Dao; R. Takahashi, The dimension of a subcategory of modules, Forum Math. Sigma 3 (2015), e19, 31 pp.
[3] Y. Yoshino, Cohen-Macaulay modules over Cohen-Macaulay rings, London Mathematical Society Lecture Note Series, 146, Cambridge University Press, Cambridge, 1990.

10:30-11:30 / 13:00-14:00

Michel Van den Bergh (Hasselt)  Noncommutative crepant resolutions and modules of covariants I & II
 Abstract  The Cohen-Macaulay modules that enter in the classical McKay correspondence are constructed from the representations of a finite group. They are known as "modules of covariants".

In the lectures I will discuss modules of covariants for reductive groups. I will start by explaining some of my own old work which gives a sufficient combinatorial condition for a module of covariants to be Cohen-Macaulay.  Then I will discuss recent joint work with Špela Špenko in which we show that such modules of covariants may be used to construct noncommutative (crepant) resolutions for quotient singularities. Finally I will discuss the the interesting special case of determinantal varieties for (skew) symmetric matrices.
 References [1] R. Buchweitz, G. J. Leuschke, and M. Van den Bergh, Non-commutative desingularization of determinantal varieties I, Invent. Math. 182 (2010), 47-115.
[2] R. Buchweitz, G. J. Leuschke, and M. Van den Bergh, Non-commutative desingularization of determinantal varieties, II, to appear in Int. Math. Res. Not.
[3] Michel Brion, Sur les modules de covariants, Annales scientifiques de l'École Normale Supérieure 26 (1993), 1-21
[4] S. Spenko and M. Van den Bergh, Non-commutative resolutions of quotient singularities, submitted.
[5] S. Spenko and M. Van den Bergh, Comparing the commutative and non-commutative resolutions for determinantal varieties of skew symmetric and symmetric matrices, submitted.
[6] M. Van den Bergh, Cohen-Macaulayness of modules of covariants, Invent. Math. 106 (1991), 389-409.


 Hai Long Dao (Kansas)  On rigid MCM modules
 Abstract  Let R be a Cohen-Macaulay local ring. A maximal Cohen-Macaulay R-module M is called rigid if Ext^1(M,M)=0. Rigid MCM modules appear naturally in many contexts, especially  divisor class groups and non-commutative resolutions of dimension 3 and higher singularities. However, even their existence remains rather mysterious. In this talk, I will describe recent progress in understanding rigid MCM modules. Some of the materials come from joint work with Ian Shipman. 
[1] H. DaoRemarks on non-commutative crepant resolutions of complete intersections. Advances in Mathematics, 224, 1021-1030. 
[2] H. Dao, Picard groups of punctured spectra of dimension three local hypersurfaces are torsion-free. Compositio Mathematica, 148, 145-152.
[3] H. Dao, I. ShipmanRepresentation schemes and rigid maximal Cohen- Macaulay modules. Selecta Mathematica, to appear. 
[4] O. Iyama, Y. Yoshino, Mutations in triangulated categories and rigid Cohen-Macaulay modules, Invent. Math. 172 (2008) 117-168.


 Osamu Iyama (Nagoya)  Higher preprojective algebras and Cohen-Macaulay representations
 Abstract  It is well-known that simple singularities R in dimension two and the path algebras kQ of (extended) Dynkin quivers have similar representation theory [1,2,3]. This is explained by using preprojective algebras, which contain both R and kQ as subalgebras.
Recently this classical result was generalized by using higher preprojective algebras and cluster tilting modules [4,5]. If time permits, I will explain some of basic results for cluster tilting based on [5].
 References [1] M. Auslander, I. Reiten, Almost split sequences for rational double points. Trans. Amer. Math. Soc. 302 (1987), no. 1, 87–97.
[2] W. Geigle, H. Lenzing,  A class of weighted projective curves arising in representation theory of finite-dimensional algebras.
Singularities, representation of algebras, and vector bundles (Lambrecht, 1985), 265–297, Lecture Notes in Math., 1273, Springer, Berlin, 1987.
[3] H. Kajiura, K. Saito, A.Takahashi, Matrix factorization and representations of quivers. II. Type ADE case. Adv. Math. 211 (2007), no. 1, 327–362.
[4] C. Amiot, O. Iyama, I. Reiten, Stable categories of Cohen-Macaulay modules and cluster categories, Amer. J. Math. 137 (2015), no. 3, 813–857.
[5] L. de Thanhoffer de V\"olcsey and M. Van den Bergh, Explicit models for some stable categories of maximal Cohen-Macaulay modules,
to appear in Mathematical Research Letters, arXiv:1006.2021.
[6] O. Iyama, M. Wemyss, Maximal modifications and Auslander-Reiten duality for non-isolated singularities, Invent. Math. 197 (2014), no. 3, 521–586.

15th June 9:30-10:30

 Izuru Mori (Shizuoka)  McKay correspondence and Beilinson correspondence for AS-regular algebras
 Abstract  McKay correspondence claims that the minimal resolution of a quotient singularity is derived equivalent to the preprojective algebra of a McKay quiver, and Beilinson correspondence claims that the projective space is derived equivalent to the path algebra of a Beilinson quiver with commutative relations.  In this talk, by integrating these ideas, we will show that a certain combination of McKay correspondence and Beilinson correspondence holds for AS-regular algebras, which are noncommutative generalizations of polynomial algebras.

[1] I. Mori, McKay-type correspondence for AS-regular algebras, J. Lond. Math. Soc. 88 (2013) 97-117.
[2] I. Mori and K. Ueyama, Ample group action on AS-regular algebras and noncommutative graded isolated singularities, Trans. Amer. Math. Soc. 368 (2016) 7359-7383.
[3] I. Mori and K. Ueyama, Stable categories of graded maximal Cohen-Macaulay modules over noncommutative quotient singularities, Adv. Math. 297 (2016) 54-92.


 Shinnosuke Okawa (Osaka)  On noncommutative Hirzebruch surfaces
Hirzebruch surfaces form a basic and important class of algebraic surfaces in commutative algebraic geometry. By definition, they are projective bundles of fiber dimension 1 over the projective line. It is well known that isomorphism classes of such surfaces are classified by non-negative integers d and that those with the same parity are connected by unobstructed deformations.
Flat deformations of the abelian category of coherent sheaves on Hirzebruch surface, or noncommutative deformations, have been studied by several people. Contrary to the case of del Pezzo surfaces, nc eformations of the d-th Hirzebruch surface are obstructed if d > 3. On the other hand, Michel Van den Bergh introduced the notion of sheaf Z-algebras and proved that any noncommutative deformation of a Hirzebruch surface over a complete Noetherian local ring is obtained from a sheaf Z-algebra associated to a locally sheaf bimodule of rank 2 on the projective line.
In this talk, I will give some introduction to this subject and explain our result on the geometric classification of those locally sheaf bimodules. I will also explain some results from the point of view of derived categories.
The talk will be based on my joint work with Izuru Mori and Kazushi Ueda.
 References [1] Michel Van den Bergh, Non-commutative P1-bundles over commutative schemes, Trans. Amer. Math. Soc. 364 (2012), no. 12, 6279–6313.
[2] Michel Van den Bergh, Noncommutative quadrics, International Mathematics Research Notices. IMRN (2011), no. 17, 3983–4026.
[3] Tarig Abdelgadir, Shinnosuke Okawa, and Kazushi Ueda, Compact moduli of noncommutative projective planes, arXiv:1411.7770.

16th June 9:30-11:00

 Ken-ichi Yoshida (Nihon)   Introduction to Ulrich modules and ideals
 Abstract  In this talk, we explain a brief history of classical Ulrich modules defined by Ulrich [U].
Definition 1 (cf. [U]). Let (A, m) be a Cohen-Macaulay local ring. Then a finitely generated A-module M is called an Ulrich A-module if it is a maximal Cohen-Macaulay A-module (that is, dim M = dim A) and $\mu_A(M)=e_m^0(M)$, where $\mu_A(M)$ denotes the number of the minimal set of genrators of M and $e_m^0(M)$ denotes the multiplicity of M with respect to m.

Ulrich gave a conjecture: any Cohen-Macaulay local ring admits an Ulrich module. This conjecture is still open.
See e.g. [BHU, Ha, HKuh, HUB] for partial answers. Moreover, Nakajima and I studied all behaviors of Ulrich modules over cyclic singularity of dim A=2. See the next talk for more details.

In 2013, Shiro Goto [GOTWY1] generalized the notion of Ulrich modules as follows:

Definiton 2 (cf. [GOTWY1]) Let (A, m) be a Cohen-Macaulay local ring and I an m-primary ideal.
A finitely generated A-module M is called an Ulrich module with respect to I if it is a maximal Cohen-Macaulay A-module such that $\ell_A(M/IM)=e_{I}^0(M)$ holds and M/IM is A/I-free.

Moreover, Shiro Goto introduced the notion of Ulrich ideal, which is a special class of "good ideals''; see [GIW, OWY1, OWY2, OWY3].

Definiton 3 (cf. [GOTWY1] Let I ⊂ A be an m-primary ideal in a Cohen-Macaulay local ring. I is said to be an
Ulrich ideal if I^2=QI for some minimal reduction Q of I and I/I^2 is A/I-free.

In the latter of the talk, the speaker gave a complete list of all Ulrich ideals and all Ulrich modules over any 2-dimensional rational double point from [GOTWY2] as the main theorem in this talk. Such a result will be generalized for simple hypersurface singularities of any dimension; see [GOTWY1, GOTWY2, GOTWY3].

Recently, Goto (see e.g. [GTT]) introduced the notion of almost Gorenstein local/graded rings using non-maximal Ulrich modules. So the significance of Ulrich modules will increase.
 References Beamer file
 [BHU] J. Brennan, J. Herzog, and B. Ulrich, Maximally generated Cohen-Macaulay modules, Math. Scand. 61 (1987), 181--203.
[GOTWY1] S. Goto, K. Ozeki, R. Takahashi, K.-i. Watanabe and K. Yoshida, Ulrich ideals and modules, Math. Proc. Cambridge Philos. Soc. 156 (2014), 137--166.
[GOTWY2] S. Goto, K. Ozeki, R. Takahashi, K.-i. Watanabe and K. Yoshida, Ulrich ideals and modules over two-dimensional rational singularities, to appear in Nagoya Math. J., arXiv:1307.2093.
[GOTWY3] S. Goto, K. Ozeki, R. Takahashi, K.-i. Watanabe and K. Yoshida, Ulrich ideals and modules for simple singularities, in preparation.
[Ha] D.~Hanes, On the Cohen-Macaulay modules of graded subrings, Trans. Amer. Math. Soc. 357 (2004) 735--756.
[HKuh] J. Herzog and M. Kuehl, Maximal Cohen-Macaulay modules over Gorenstein rings and Bourbaki-sequences, Commutative algebra and combinatorics (Kyoto, 1985), 65--92, Adv. Stud. Pure Math., 11,
North-Holland, Amsterdam, 1987.
[HUB] J. Herzog, B. Ulrich and J. Backelin, Linear maximal Cohen-Macaulay modules over strict complete intersections, J. Pure Appl. Algebra, 71 (1991), 187--202.
[GIW] S. Goto, S.-i. Iai, and K.-i. Watanabe, Good ideals in Gorenstein local rings, Trans. Amer. Math. Soc. 353 (2000), 2309--2346.
[GTT] S. Goto, R. Takahashi and N. Taniguchi, Almost Gorenstein rings -towards a theory of higher dimension, J. Pure Appl. Algebra, 219 (2015), 2666--2712.
[NY] Y. Nakajima and K. Yoshida, Ulrich modules over cyclic quotient surface singularities, preprint, 2015
[OWY1] T.~Okuma, K.-i. Watanabe, and K. Yoshida, Good ideals and $p_g$-ideals in two-dimensional normal singularities, to appear in Manuscripta Mathematica, 2016 (available from arXiv:1407.1590).
[OWY2] T.~Okuma, K.-i. Watanabe, and K. Yoshida, Rees algebras and $p_g$-ideals in a two-dimensional normal local domain, available from arXiv:1511.00827.
[OWY3] T.~Okuma, K.-i. Watanabe, and K. Yoshida, A characterization of two-dimensional rational singularities
via Core of ideals, available from arXiv:1511.01553.


 Yusuke Nakajima (Nagoya)   Ulrich modules over quotient surface singularities
 Abstract  An Ulrich module is a maximal Cohen-Macaulay module which has the maximum number of generators. In this talk, I will discuss Ulrich modules over quotient surface singularities. In the process of finding them, the notion of special Cohen-Macaulay modules (see [Wun1,Wun2]) plays a crucial role. Especially, I will observe how to classify Ulrich modules over cyclic quotient surface singularities by using special ones.
This talk is based on a joint work with Ken-ichi Yoshida [NY].
 References  [NY] Y.Nakajima and K. Yoshida, Ulrich modules over cyclic quotient surface singularities, arXiv:1504.07688
[Wun1] J. Wunram, Reflexive modules on cyclic quotient surface singularities, Lecture Notes in Mathematics, Springer-Verlag 1273 (1987), 221--231.
[Wun2] J. Wunram, Reflexive modules on quotient surface singularities, Math. Ann. 279 (1988), no. 4, 583--598.


 Robin Hartshorne
(UC Berkeley)
 Ulrich Bundles on Cubic Hypersurfaces
 Abstract  I will review definitions, properties and existence questions for ACM bundles and Ulrich bundles on projective varieties. Then I will discuss in some detail the Ulrich bundles on cubic hypersurfaces in P^3 and P^4.
 References [1] Casanellas, Hartshorne, ACM bundles on cubic surfaces, J Eur. math. Soc. 13, 2004 709-731
[2] Casanellas, Hartshorne, Geiss, Schreyer, Stable Ulrich Bundles, , Internat. J. Math. 23, 2012, no. 8, 50 pp.


 Iku Nakamura (Hokkaido) Global McKay correspondence for quotient surface singulariteis (based on a joint work with A. Ishii)
 Abstract This is a report of our joint work in progress. The G-Hilbert scheme here is a scheme which parametrizes -dimensional G-invariant subschemes (each called a G-cluster) of C^2 of length the order of G. Two dimensional McKay correspondence for quotient singularities (in terms od G-Hilbert schemes [Ito-Nakamura99] and [Ishii02]) is re-formulated on the G-hilbert scheme as a global isomorphism of certain universal coherent sheaf with the sume of O_E(-1) for E all irreducible components of the exceptional sent. The quiver structure in the structure sheaf of each cluster is discussed.
 References [Ito-Nakamura99] Hilbert Schemes and simple singularities, New Trends in Algebraic Geometry, Proc. of EuroConference on Algebraic Geometry, Warwick 1996, ed. by K. Hulek et al., CUP, (1999), 151-233.
[Ishii02] Ishii, A., On the McKay correspondence for a finite small subgroup of GL(2,C). J. Reine Angew. Math. 549 (2002), 221–233.
[Ishii-Nakamura] in preparation


 Poster session  Posters will be shown from Wednesday to Friday at the room 110
 Presenters Alvaro Nolla de Celis (Rey Juan Carlos): Group actions and dimer models,
Seung-Jo Jung (KIAS): On the Craw--Ishii conjecture,
Joseph Karmazyn (Bath): Noncommutative Knörrer periodicity for cyclic quotient surface singularities,
Antoine Caradot (Lyon): Inhomogeneous Kleinian singularities and quivers,
Tomoaki Shirato (Nagoya): On Frobenius split Abelian fiber spaces over curves,
Kenta Sato (Tokyo): Stability of test ideals of divisors with small multiplicity,
Chihiro Enomoto & Ken-Ichi Yoshida (Nihon): Cleannes of Cohen-Macaulay monomial ideal generated by five elements,
Wahei Hara (Waseda): Rouquier Dimension and Orlov Spectrum of Singular Varieties,
Ben Wormleighton (Berkeley): G-Hilb for trihedral groups,
Genki Ouchi (Tokyo): Automorphisms of positive entropy on some hyperKahler manifolds via derived automorphisms of K3 surfaces.

*** 18:00- Conference dinner at Kyoto Univ. Coop restaurant ***

17th June 9:30-11:00

 Stefan Schröer
(Heinrich Heine Universität)
 Wild quotient singularities
 Abstract   I give a general introduction to wild quotient singularities, with an emphasis on the actions of Z/pZ on power series ring of characteristic p.  I will start with an historical overview, and also touch modular representation theory, and finally describe some joint work with Dino Lorenzini.
 References [1] M. Artin: Wildly ramified $Z/2$ actions in dimension two. Proc. Amer. Math. Soc. 52 (1975), 60--64.
[2] H. Ito, S. Schr\"oer: Wildly Ramified Actions and Surfaces of General Type Arising from Artin--Schreier Curves. In: C. Faber, G. Farkas and R. de Jong (eds.), Geometry and arithmetic, pp. 213--241. Eur. Math. Soc., Z\"urich, 2012.
[3] H. Ito, S. Schr\"oer: Wild quotient surface singularities whose dual graphs are not star-shaped. Asian J. Math. 19 (2015), 951--986.
[4] D. Lorenzini: Wild quotient singularities of surfaces. Math. Z. 275 (2013),  211–-232.
[5] D. Lorenzini, S. Schr\"oer: Moderately ramified  actions and their rings of invariants in positive characteristics. Work in progress.
[6] B. Peskin: Quotient-singularities and wild $p$-cyclic actions. J. Algebra  81  (1983), 72--99.
[7] M. Romagny: Effective models of group schemes. J. Algebraic Geom. 21 (2012), 643--682.


 Shuji Saito (Tokyo I. Tech.)  McKay principle and Hasse princible
 Abstract  Quite a few examples have been observed which show that an arithmetic method can play a significant role for a geometric question. In this talk we present such an example. On the geometric side the McKay principle is formulated for the homotopy type of the dual complex of the exceptional divisors of a resolution of a quotient singularity. To prove it, we use the cohomological Hasse principle, which is a higher dimensional generalization of the global-local principle for division algebras over number fields, a classical theorem in number theory. This is a joint work with M. Kerz.


 Takehiko Yasuda (Osaka)  Computational aspects of the wild McKay correspondence
 Abstract  I will talk about interesting computations appearing in the wild McKay correspondence, a certain generalization of the McKay correspondence to arbitrary characteristic. This version of the McKay correspondence roughly says that a stringy invariant of a quotient singularity is equal to a weighted count of Galois extensions of a local field (power series fields or p-adic fields). It is a fun to observe how phenomena for local fields and ones for singularities correspond one another through the McKay correspondence. I will present concrete computations realizing such coincidences as:
1. Bhargava's mass formula vs. Ellingsrud-Stromme's generating function of the Hilbert scheme of points
2. Convergence/divergence of weighted counts of Galois extensions vs. the property of singularities' (not) being log terminal
3. Dualities in mass formulas vs. the Poincare duality and equisingularities
 References [1] T. Yasuda, The wild McKay correspondence via motivic integration (Japanese), to appear in Sugaku
[2] M.M. Wood and T. Yasuda, Mass formulas for local Galois representations and quotient singularities II: dualities and resolution of singularities, arXiv:1505.07577.
[3] T. Yasuda, The wild McKay correspondence and p-adic measures, arXiv:1412.5260, to appear in Journal of the European Mathematical Society.
[4] T. Yasuda, Wilder McKay correspondences, Nagoya Mathematical Journal, Volume  221, Issue 01,  pp 111 - 164 
[5] M.M. Wood and T. Yasuda, Mass formulas for local Galois representations and quotient singularities I: a comparison of counting functions, Int Math Res Notices, Volume 2015, Issue 23, pp 12590-12619.
[6] T. Yasuda, Toward motivic integration over wild Deligne-Mumford stacks, arXiv:1302.2982, to appear in the proceedings of the conference "Higher Dimensional Algebraic Geometry - in honour of Professor Yujiro Kawamata's sixtieth birthday"
[7] T. Yasuda, The p-cyclic McKay correspondence via motivic integration, Compositio Mathematica, 150, 1125-1168.