Geometric McKay Correspondance

Resumen:

The classical McKay correspondence gives a one to one correspondence between irreducible components of the exceptional divisor of the minimal resolution of the singularities ADE with irreducible representations of the finite subgroups of SU(2). 

We are more interested in a geometric interpretation, i.e., How can we reconstruct the dual graph of the minimal resolution of a singularity ADE from the irreducible representations of a finite subgroup of SU(2) completely in geometrical terms? This is the geometric McKay correspondence and it was first answered by Artin and Verdier, they gave a one to one correspondence between irreducible components of the exceptional divisor and isomorphism classes of indecomposable reflexive modules. Later Esnault and Knörrer gave the so called multiplication formula which allow us to recover the incidence between exceptional divisors.

Further work of Esnault, Wunram and Kahn went deeper into the subject. Wunram generalized this to rational singularities, giving a one to one correspondence between irreducible components of the exceptional divisor of the minimal resolution and isomorphism classes of certain class of reflexive modules. Later Kahn generalized some constructions to the general case of complex surface singularities and he studied reflexive modules for some kind of non rational singularities.

We will focus on the ideas and proofs of this results and if the time allows us, we will give some generalizations which are joint work with Agustín Romano and José Luis Cisneros.

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