The 12th International Conference
by Graduate School of Mathematics, Nagoya University

Conference on resolution of singularities and the McKay correspondence

Bldg. Sci. 1(理学部1号館) Room 309/509
Graduate School of Mathematics, Nagoya University
Organizer: Yukari Ito (Nagoya University)
1 May  2 May 3 May
 Room 309  309(AM)/509(PM) Room 509 
Yukari Ito (Nagoya) 
Takehiko Yasuda (Osaka)
Kazushi Ueda (Osaka)  
Alastair Craw 1 (Glasgow)
Michael Wemyss (Edinburgh)
Hokuto Uehara (TMU) 
Timothy Logvinenko (Warwick)
Martin Herschend (Nagoya)
Laurent Demonet (Nagoya)
Yuhi Sekiya (Nagoya)
Osamu Iyama (Nagoya)
Hiroyuki Minamoto (Nagoya)
Alastair Craw 2 (Glasgow)
Alvaro Nolla de Celis (UNIR)
Akira Ishii (Hiroshima)
Miles Reid (Warwick/Sogang)

<Titles and Abstracts>

Yukari Ito (Nagoya) Introduction to the McKay correspondence
I will talk on existence of crepant resolutions and the McKay correspondence in dimension three and higher.

Alasatair Craw 1 (Glasgow) Reid's recipe: the past
In his well known (though somewhat speculative) Kinosaki preprint from 1996, Miles Reid introduced a combinatorial cookery with characters of certain finite abelian subgroups G of SL(3,C) to construct a basis of the integral cohomology of the G-Hilbert scheme in each of his examples. I will give an introductory and opinionated talk about this recipe and its proof, with the intention of giving the audience several views on this classical version of Reid's recipe.

Timothy Logvinenko (Warwick) Derived Reid's recipe for abelian subgroups of SL3(C)
Derived Reid's recipe is a correspodence where we use BKR derived category equivalence to assign to every irreducible representation of a finite subgroup of SL_3(C) a subvariety of the exceptional set of the crepant resolution G-Hilb C^3 of the quotient singularity C^3/G. This generalises the classical McKay correspondence for subgroups of SL_2(C) which assigns to each representation an irreducible exceptional divisor. I discuss a recent work with Alastair Craw and Sabin Cautis in which we compute derived Reid's recipe for all abelian subgroups of SL3(C). A new notion of a "CT-subdivision" allows for a concise demonstration of the ideas at the heart of our main theorem."

Yuhi Sekiya (Nagoya) McKay correspondence and essential representations
Special McKay correspondence says that, in the two dimensional case, there is an one-to-one correspondence between exceptional curves and special representations. First I show examples that the minimal resolution is constructed as a moduli space by using only special representation. In this viewpoint, I try to extend this result to higher dimensional case.

Alastair Craw 2 (Glasgow) Reid's recipe: the future
I will discuss work in progress on a generalisation of Reid's recipe that goes well beyond the original context envisaged by Reid. The categorical viewpoint taken here is inspired by work of Cautis-Logvinenko and subsequent work of Logvinenko.

Akira Ishii (Hiroshima) Dimer models and crepant resolutions
We consider variation of moduli spaces of the quiver representations associated with dimer models and apply it to a problem on derived equivalences.

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Takehiko Yasuda (Osaka) Motivic integration and wild group actions
The cohomological McKay correspondence proved by Batyrev is the equality of an orbifold invariant and a stringy invariant. The former is an invariant of a smooth variety with a finite group action and the latter is an invariant of its quotient variety. Denef and Loeser gave an alternative proof of it which uses the motivic integration theory developped by themselves. Then I pushed forward with their study by generalizing the motivic integration to Deligne-Mumford stacks and reformulating the cohomological McKay correspondence from the viewpoint of the birational geometry of stacks. However all of these are about tame group actions (the order of a group is not divisible by the characteristic of the base field), and the wild (= not tame) case has remained unexplored.
In this talk, I will explain my attempt to examine the simplest situation of the wild case. Namely linear actions of a cyclic group of order equal to the characteristic of the base field are treated. A remarkable new phenomenon is that the space of generalized arcs is a fibration over an infinite dimensional space with infinite dimensional fibers, where the base space is the space of Artin-Schreier extensions of $k((t))$, the field of Laurent series.

Michael Wemyss (Edinburgh) Simultaneous resolution of non-ADE surfaces, and relationship to 3-folds.
In ADE surface singularities, it is well-known that the versal deformation space admits a simultaneous resolution after base change. For non-ADE surfaces, there are now many components in the versal space, and only one (namely the Artin component) admits a simultaneous resolution. I will explain how to achieve this, at least in Type A, using reconstruction algebra techniques. This has applications in higher dimensional birational geometry, which I will also explain.

Martin Herschend (Nagoya) n-representation infinite algebras constructed from skew-group algebras
This talk is based on joint work with Osamu Iyama and Steffen Oppermann. I will introduce the class of n-representation infinite algebras, which are an analogue of hereditary representation infinite algebras from the viewpoint of higher dimensional Auslander-Reiten theory. These algebras can be characterized by the fact that their higher preprojective algebras are (n+1)-Calabi-Yau. Skew-group algebras of the polynomial algebra by finite subgroups H of SL_{n+1} are a source of (n+1)-Calabi-Yau algebras. I will explain how we can realize such skew-group algebras as preprojective algebras of n-representation infinite algebras for the case when H is abelian.

Osamu Iyama (Nagoya) Algebraic McKay correspondence for n-representation infinite algebras
n-representation infinite algebras A give a distinguished class of finite dimensional algebras of global dimension n. They bijectively correspond to bimodule (n+1)-Calabi-Yau algebras B of Gorenstein parameter 1 (Amiot-I-Reiten, Keller, Minamoto-Mori). For a nice idempotent e, the algebra C:=eBe is a (non-commutative) Gorenstein algebra, and the derived/cluster category of A/e is triangle equivalent to the stable category of (graded) Cohen-Macaulay C-modules. This gives a higher dimensional generalization of Auslander's algebraic McKay correpondence. This talk is based on joint works with Amiot-Reiten and Herschend-Oppermann.

Alvaro Nolla de Celis (UNIR) On quotients by non-Abelian intransitive groups in SL(3,C) and their resolutions
Intransitive subgroups in SL(3,C) are groups isomorphic to finite subgroups of GL(2,C), and they constitute a nice family to test several aspects of the McKay correspondence with non-Abelian groups. I will introduce this class of subgroups and give the ingredients to calculate G-Hilb in this cases.

Miles Reid (Warwick/Sogang) Crepant resolutions and A-Hilb for 1/r(a,b,1,1,..,1)
An easy n-dimensional result says (in the coprime case) that 1/r(a,b,1,1,..,1) has a crepant resolution resolution if and only if the point nearest the (x1=0) face is (1,r-(n-2)c-1,c,.. c) (c repeated n-2 times) with (n-2)c - 1 < r, and every entry of the Hirzebruch-Jung continued fraction of r/(r-(n-2)c-1) is == 2 mod n-2. The calculation of A-Hilb in these cases by an algorithm in the style of Nakamura and Craw-Reid is also interesting, although it gets somewhat trickier as n gets larger.

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Kazushi Ueda (Osaka) On orbifold projective lines and their higer-dimensional generalizations
We first recall the theory of weighted projective lines in the sense of Geigle and Lenzing, and its application to McKay correspondence.
Then we discuss generalizations to higher dimensions.

Hokuto Uehara (TMU) Frobenius morphisms and derived categories on two dimensional toric Deligne--Mumford stacks
For toric Deligne--Mumford stacks X, we can consider a cerrtain generalization of Frobenius endomorphism. For such an endomorphism on 2-dimensional toric Deligne--Mumford stacks X, we show that the push-forward of the structure sheaf generates the bounded derived category of coherent sheaves on X.
My talk is based on a joint work with Ryo Ohkawa.

Laurent Demonet (Nagoya) Quiver of skew-group algebras. Application to hereditary and preprojective algebras
In this talk, we shall present a formula to compute the quiver of a skew-group algebra of an algebra given by a quiver and relations by the action of a semi-simple finite group. As an application, we compute the skew-group algebras of hereditary and preprojective algebras (in good cases), the later one generalizing the usual formula of McKay correspondence. We also explain how to generalize these kind of result to a more abstract setting (mainly the case of $d$-Calabi-Yau dg-algebras).

Hiroyuki Minamoto (Nagoya) Behavior of Serre functors and skew group rings