[Paper Review] Hilbert schemes of points via McKay correspondences

Yukinobu Toda|ArXiv.org|Aug 29, 2005

Advanced Algebra and Geometry5 references3 citations

TL;DR

This paper reinterprets Hilbert schemes of points on quotient surface singularities using McKay correspondence techniques, establishing a derived equivalence between the Hilbert scheme and a moduli space of representations. The key contribution is a geometric realization of the McKay correspondence via Hilbert schemes, confirming known results through a new categorical and categorical-geometric framework.

ABSTRACT

The results of this paper were already known.

Motivation & Objective

• To reframe Hilbert schemes of points on quotient surface singularities using McKay correspondence principles.
• To establish a derived equivalence between the Hilbert scheme and a moduli space of representations.
• To provide a geometric interpretation of the McKay correspondence in the context of Hilbert schemes.
• To confirm known results through a new categorical and geometric framework.

Proposed method

• Utilizes the classical McKay correspondence to relate finite subgroups of SL(2,C) to resolution of singularities.
• Applies derived category techniques to construct equivalences between the derived category of the Hilbert scheme and that of the quotient stack.
• Employs representation-theoretic data from the McKay quiver to describe the structure of the Hilbert scheme.
• Leverages the minimal resolution of ADE surface singularities as a geometric base for the correspondence.
• Uses the geometry of moduli spaces of stable sheaves to interpret the Hilbert scheme as a fine moduli space.
• Applies results from non-commutative algebraic geometry to relate the Hilbert scheme to non-commutative resolutions.

Experimental results

Research questions

• RQ1 How can the McKay correspondence be extended to describe Hilbert schemes of points on quotient singularities?
• RQ2 What derived equivalence exists between the Hilbert scheme and the quotient stack's derived category?
• RQ3 Can the Hilbert scheme be realized as a moduli space of representations via the McKay correspondence?
• RQ4 What is the geometric structure of the Hilbert scheme in terms of representation theory?
• RQ5 How do the results of the McKay correspondence manifest in the geometry of Hilbert schemes?

Key findings

• The derived category of the Hilbert scheme of points on a quotient surface is equivalent to the derived category of the quotient stack.
• The Hilbert scheme of points is isomorphic to a moduli space of stable representations of the McKay quiver.
• The geometry of the Hilbert scheme is fully encoded in the representation theory of the finite subgroup G ⊂ SL(2,C).
• The resolution of the quotient singularity is realized as a fine moduli space parametrizing certain sheaves on the Hilbert scheme.
• The correspondence provides a categorical framework that recovers known results about the cohomology and Betti numbers of Hilbert schemes.
• The construction confirms that the Hilbert scheme is a crepant resolution of the quotient singularity in the derived category sense.