Seminar on the McKay correspondence

SEMINAR ON THE McKAY CORRESPONDENCE

8-10 March 2010
Graduate School of Mathematics
Nagoya University

SCHEDULE

2010/3/8 (Monday)
10:30-12:00 Timothy Logvinenko (MPIM Bonn & Warwick) I "The McKay correspondence: from geometrical to derived and back again".
13:30-14:30 Toshihiro Hayashi (Nagoya) "Examples of crepant resolutions of 4-dimensional non-abelian quotient singularities".
15:00-16:30 Michael Wemyss (Oxford) I "Mutations of Quivers in the Minimal Model Program".
17:00-18:30 Akira Ishii (Hiroshima) I "Special McKay correspondence and exceptional collections".

2010/3/9 (Tuesday)
10:00-11:30 Timothy Logvinenko (MPIM Bonn & Warwick) II "On a sufficiency criterion for the derived McKay equivalences".
13:00-14:00 Alvaro Nolla de Celis (Nagoya) "On Hilb of Hilb".
14:30-16:00 Michael Wemyss (Oxford) II "Mutations of Quivers in the Minimal Model Program".
16:30-18:00 Akira Ishii (Hiroshima) II "Dimer models and exceptional collections".

2010/3/10 (Wednesday)
10:30-12:00 Miles Reid (Sogang, Seoul & Warwick) "Godeaux surfaces".

All talks will be at Room 409.

ABSTRACTS

Michael Wemyss (I,II)
Title: Mutations of Quivers in the Minimal Model Program.
Abstract: Following work of Bridgeland in the smooth case and Chen in the terminal singularities case, I will explain a proposal that extends the existence of flops for threefolds (and the required derived equivalences) to also cover canonical singularities. Moreover this technique conjecturally says much more than just the existence of the flop, as it shows how the dual graph changes under the flop and also which curves in the flopped variety contract to points without contracting divisors. This allows us to continue the Minimal Model Programme on the flopped variety in an easy way, thus producing many varieties birational to the given input.
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Timothy Logvinenko I
Title: The McKay correspondence: from geometrical to derived and back again.
Abstract: In this talk I introduce the classical geometrical McKay correspondence for the finite groups of SL_2(C), explain the K-theoretic construction of it by Gonzales-Sprinberg and Verdier and how that leads to the derived McKay equivalence of Bridgeland, King and Reid. I then show how the two-dimensional geometrical McKay correspondence can be recovered from this derived equivalence and how this generalises to the dimension three.
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Timothy Logvinenko II
Title: On a sufficiency criterion for the derived McKay equivalences.
Abstract: In their groundbreaking work Bridgeland, King and Reid have employed the Intersection Theorem from commutative algebra to reduce sufficiency criteria for the derived McKay equivalence on G-Hilb(C^n) to a certain condition on the dimension of the exceptional fiber. This condition is satisfied automatically in dimensions 2 & 3 - and is almost never satisfied in dimensions 4 and above. In this talk I show that some further exploitation of the Intersection Theorem allows to establish a much finer sufficiency criterion which works for an arbitrary family of G-Constellations and could realistically be applied in higher dimensions, in particular - in dimension 4.
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Akira Ishii I
Title: Special McKay correspondence and exceptional collections.
Abstract: For a finite subgroup G of SL(2, C), the original McKay correspondence establishes a bijection between non-trivial irreducible representations of G and irreducible exceptional curves of the minimal resolution of the quotient singularity C^2/G. In the case of a finite small subgroup G of GL(2, C), there are in general more irreducible representations than exceptional curves. Wunram and Riemenschneider defined "special" representations of G, which correspond to exceptional curves. In this talk, I would like to explain an semi-ortogonal decomposition of the G-equivariant derived category of C^2 into "special" and "non-special" parts, where the special part corresponds to the minimal resolution and the non-special part is generated by an exceptional collection. This is a joint work with Kazushi Ueda.
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Akira Ishii II
Title: Dimer models and exceptional collections.
Abstract: A dimer model is a bicolored graph on a torus, which encodes an information of a quiver with relations. In some special cases, the associated quiver with relations is the McKay quiver for a finite abelian subgroup of SL(3, C). If the dimer model is "consistent", then there is a derived equivalence between the associated quiver with relations and a crepant resolution of a three dimensional toric Gorenstein singularity. By using this fact, we can construct a full strong exceptional collection consisting of line bundles on an arbitrary two-dimensional smooth toric weak Fano stack. A description of the total endomorphism algebra is given in terms of dimer models, which should be relevant to describing the homological mirror symmetry for such a stack. This is also a joint work with Kazushi Ueda.