不変元についての注意(A. 理学)

Let V be an affine variety and let G be a connected linear algebraic group acting on V in the usual sense. Let R=K[f_1,f_2,…, f_n] be a coordinate ring of V. Then our main result is that : When G acts rationally, an element f of R is G-invariant if and only if ƒ is B-invariant with a suitable Borel sub group B of G. Let V be a surface, and let V\u27 be a non-singular surface which is birationally equivalent to V and dominates V. We denote by T the anti-regular map from V onto V\u27. If V\u27 satisfies the following conditions 1)∿4) then we shall say that V\u27 is a resolved surface of V. Let Ω^* be the set of all points of V\u27 which correspond to singular points of V, then Ω^* is a closed set of V\u27. 1) T is biregular at every simple point of V. 2) Ω^* is pure of dimension one and each irreducible component of Ω^* is non-singular. 3) Two components of Ω^* make a normal crossing at any common point. 4) No three components of Ω^* have a common points. Theorem 1. A resolved surface V\u27 of a given surface V exists. Theorem 2. Let F be the set of all resolved surfaces of V. If F has two elements V_1 and V_2,then it has a third element V_3 which dominates V_1 and V_2