mckay correspondence iku nakamura

32 IKU NAKAMURAThen we see p 1 B 1 − ω 2 p 2 B 2 + ωp 3 B 3 = ω(1 − ω)(ψ 2 − ωuϕ). Hence ψ 2 −ωuϕ ∈ I univ , whence ψ 3 −ωuW ∈ I univ . Similarly we see p 1 B 1 −p 2 B 2 +p 3 B 3 =(ω − 1)sW − 3tT, which belongs to I univ . We also have (1 − ω 2 ){(1 + s)ϕ 2 −ωuψ} = p 1 B 1 − ωp 2 B 2 + ω 2 p 3 B 3 ∈ I univ . Hence (1 + s)ϕ 3 − ωuW ∈ I univ .From π ∗ m ∩ I univ = {0} it follows thatIt follows thatπ ∗ ψ 3 = ωuπ ∗ W, (1 + s)π ∗ ϕ 3 = ωuπ ∗ W,(ω − 1)sπ ∗ W =3tπ ∗ T,π ∗ ϕ 3 − π ∗ ψ 3 = 3(2ω +1)π ∗ T 2 .π ∗ W = ω2 u 21+s ,π∗ ϕ 3 =u 3(1 + s) 2 ,π∗ ψ 3 = u31+s ,3π ∗ T =(1− ω 2 u 2 s)(1 + s)t , (1 − ω)su = t2 .Though the relation (1 − ω)su = t 2 looks singular at P (ρ 2 ,ρ 3 ), the points = t = 0 of Hilb G (A 2 ), it is not singular at all because s and t/s are regularparameters at P (ρ 2 ,ρ 3 ).The condition W ≠ S 1 V 5 (ρ 2 ),S 1 V 5 (ρ ′ 2),S 1 V 5 (ρ ′′2) implies s(1 + s) ≠0,s ≠∞. The parameters s, t and u are related by (1 − ω)su = t 2 . Hence π ∗ m =(π ∗ T )=(u 2 /t) =(t 3 ) along E(ρ 3 ) \ Sing(E), whence E fund =3E(ρ 3 ) there.We see mod n XtB 1 = tp 1 (ϕ + ωψ)+stp 1 ϕ + t 2 p 2 p 3 + tup 1= tp 1 (ϕ + ωψ)+stp 1 ϕ + t 2 p 2 p 3 − 3ω(1 + s)p 1 T= −ω 2 (s − ω)p 3 B 2 +(s − ω 2 )p 2 B 3 ,tB 2 = ω(s − ω)p 1 B 3 − (s − ω 2 )p 3 B 1 ,tB 3 = −ω(s − ω)p 2 B 1 +(s − ω 2 )p 1 B 2 .1This proves tB i =0inV. Hence V is S[s, ]-free of rank three. Thiss(1+s)completes the proof of freeness of V over E(ρ 3 ) \ Sing(E) for E 6 .6.2.8. We explain very briefly the most complicated case of E 7 , that is, theρ 4 -case. The finite group G involved is the binary octahedral group O, andthe invariant ring of G is generated by homogeneous polynomials of degree 8,12, and 18, where we note that 18 is also the Coxeter number of E 7 .Any point I = I 4 (W ) ∈ E(ρ 4 ) \ Sing(E) is given by an irreducible G-submodule of V 9 (ρ 4 ) with W ≠ S 1 V 8 (ρ ′′2 ),S 1V 8 (ρ 3 ),S 1 V 8 (ρ ′ 3 ) under the notationof section 5. Then we seeHom S (I/I 2 , S/I) = Hom (W, V 9 (ρ 4 )/W ) ⊕ Hom (W, V 7 (ρ 4 )).The versal deformation I univ is generated over E(ρ 4 ) \ Sing(E) by five elementsB 1 ,B 2 ,B 3 ,B 4 ,A := A 8 − π ∗ (A 8 ), where A 8 = W is the same as W of