mckay correspondence iku nakamura

28 IKU NAKAMURAπ : X → A 2 /G is known by Theorem 2.5 to be the minimal resolution. Wedefine I univ to be the ideal sheaf of O 2 ×X defining Z univ in A 2 × X. Then wehave an exact sequence 0 → I univ → O 2 ×X → O Zuniv → 0.We define I X to be the defining ideal of X as a subscheme of X ≃(A 2 /G) × ( 2 /G) X of (A 2 /G) × X, n X := I X O 2 ×X andV (I univ ):=I univ /mI univ + n X ,which we denote by V. The sheaf V is a finite O 2 ⊗O X -module supported by{0}×E because mO 2 = O 2 outside the origin and {0}×X∩Supp(Z univ ) red ={0}×E. It is clear that mV = n X V = 0 from the definition of V.Let m be the maximal ideal of the unique singular point of A 2 /G. By thedefinition of n X , φ ∗ F = π ∗ F mod n X for any F ∈ m.We prove next π ∗ (m)V = 0. Let H ∈ I univ and a ∈ π ∗ m. Then there aresome F k ∈ O X and A k ∈ m such that a = ∑ k F kπ ∗ (A k ). Since n X ⊂ I univ(See subsection 3.8), we have aH ≡ ∑ ∑k F kφ ∗ (A k )H in I univ /n X . Howeverk F kφ ∗ (A k )H ∈ mI univ , which proves aH =0inV. It follows π ∗ (m)V =0.Since π ∗ m is the ideal of O X defining the fundamental cycle E fund of E, thisimplies that V is a finite O Efund -module.Since E(ρ) is a subscheme of E fund , V⊗O E(ρ) is a finite O E(ρ) -module andwe have a natural homomorphism(2)V→ ∑V⊗O E(ρ) .ρ∈Irr GWe prove this is an isomorphism in subsections 6.2 and 6.3.6.2. Freeness outside Sing(E). First we prove that (2) is an isomorphismat a nonsingular point of E. Let S = C[x, y].Let I ∈ E(ρ) \ Sing(E). Then the ideal I is generated by a nonzero irreducibleG-submodule W of V (ρ) =V h−d(ρ) (ρ)+V h+d(ρ) (ρ) and n by section 4or by [IN99], Theorem 10.7.First we consider the case D 5 .6.2.1. Let ρ = ρ 1 . Then W ≠ V 6 (ρ 1 ). As Hilb G (A 2 ) is nonsingular and twodimensional,the tangent space T [I] (Hilb G (A 2 )) is exactly two-dimensional.By subsection 4.3Hom S (I/I 2 , S/I)[ρ 0 ] = Hom (W, V 6 (ρ 1 )) ⊕ Hom (V 6 (ρ 0 ),V 0 (ρ 0 )).We note that I is generated by W and A 6 by subsection 4.9. SinceT [I] (E(ρ 1 )) = Hom (W, V 6 (ρ 1 )), the parameter t of Hom (V 6 (ρ 0 ),V 0 (ρ 0 ))gives a defining equation of E(ρ 1 ).By subsection 4.3 the ideal I univ of Z univ is over E(ρ 1 ) \ Sing(E) generatedby xy+s(x 6 −y 6 ) and A 6 +t. Since A 6 +t ∈ n X , the quotient V is S⊗C[s, t]/tfreeof rank one, hence O E -free of rank one.