DMM Seminar

Speaker: Timothy Logvinenko (Warwick)
Title: Derived Reid's recipe for threefold singularities (Part II)
Date (JST): Tue, Nov 15, 2011, 10:00 - 11:30
Place: Seminar Room A
Abstract: Part II:

Here I explain the technical concepts behind the proofs of the results discussed in Part I.

Derived Reid's recipe works by computing the images \Psi(O_0 x \chi) in D(Y) where \Psi is the inverse of the BKR equivalence, O_0 is the skyscraper at the origin of C^3 and \chi is an irreducible representation of G.

I explain how \Psi(O_0 x \chi) can be computed as the total complex of a certain skew-commutative cube of line bundles. I then explain how to compute the cohomologies of this complex in terms of which arrows of the McKay quiver vanish along which exceptional divisors in the quiver representation associated to the universal family of G-clusters. For a single divisor E this vanishing data can be represented by a "sink-source graph of E". It turns out that the possible shapes of this graph are in exact correspondence with the possible shapes of the toric fan of Y around E. It is this beautiful correspondence which lies at the heart of derived Reid's recipe.