On G/N-Hilb of N-Hilb

Abstract

Abstract. In this paper we consider the iterated G-equivariant Hilbert scheme G/N -Hilb(NHilb) and prove that G/N -Hilb(N -Hilb(C 3 )) is a crepant resolution of C 3 /G isomorphic to the moduli space M θ (Q) of θ-stable representations of the McKay quiver Q for certain stability condition θ. We provide several explicit examples to illustrate this construction. We also consider the problem of when G/N-Hilb(N-Hilb) is isomorphic to G-Hilb showing the fact that these spaces are most of the times different.

$G$-constellations and the maximal resolution of a quotient surface singularity

“…In the case of G-constellations for non-abelian G & GLð2; CÞ, we shall use the iterated construction of moduli spaces for a normal subgroup of G as in [IINdC13]. In order to do so, we have to consider G-constellations on a variety, rather than an a‰ne space.…”

“…We can show that the conjecture is true if G is abelian (Theorem 5) by using the result of [CI04]. The idea in the non-abelian case of Theorem 1 is to use iterated construction of moduli spaces as in [IINdC13] and reduce the problem to the abelian group case. Namely, let N be the cyclic group generated by ÀI , which is a normal subgroup of every non-abelian finite small subgroup.…”

Walls for G-Hilb via Reid's Recipe

“…The unlocking procedure for this curve is shown in Fig. 26 giving G-igpC 3 q t1, 2,3,4,5,6,8,9,10,12,13,14,16,17,18,20,21,22,24,25,26,28,29,30, 32, 33, 34u. Notice that every chain meeting the 3-chain in a vertex is broken there.…”

“…We complete the description of wall-crossing behaviours from [25] by calling a wall Type 0 if the corresponding contraction is a isomorphism. This explicit and malleable description of the walls for C 0 has applications to studying the birational geometry of other crepant resolutions of A 3 {G. In forthcoming work [18], the author and Y. Ito use this description of C 0 to study the geometry of another Hilbert scheme-like resolution introduced in [14] called the "iterated G-Hilbert scheme" or "Hilb of Hilb".…”

$G$-constellations and the maximal resolution of a quotient surface singularity

“…We can show that the conjecture is true if G is abelian (Theorem 5.1) by using the result of [CI04]. The idea in the nonabelian case of Theorem 1.1 is to use iterated construction of moduli spaces as in [IINdC13] and reduce the problem to the abelian group case. Namely, let N be the cyclic group generated by −I, which is a normal subgroup of every non-abelian finite small subgroup.…”