Finite generation of the Nagata invariant rings in A-D-E cases

(t1, . . . , tn) ∈ C S2n = C[x1, . . . , xn, y1, . . . , yn], { xi −→ xi yi −→ tixi + yi, The restriction of this action to a general linear subspace G ⊂ C is called an action of Nagata type. In [M], generalizing the result of Nagata [N1] (r = 3 and n = 16), we proved the infinite generation of the invariant ring S in the case where the inequality 1/2+1/(n− r)+1/r ≤ 1 holds, where r is the codimension of G. In this article, we shall show the converse: Theorem The invariant ring S of Nagata type is finitely generated if 1/2 + 1/(n− r) + 1/r > 1.