, ? is surjective, and its fibers are exactly the G-orbits in X
, Under these assumptions, the topological space Y may be identified with the orbit space X/G equipped with the quotient topology, in view of (1) and (2). Moreover, the structure of variety on Y is uniquely defined by (3) (which may be rephrased as the equality of sheaves O Y = ? * (O X ) G ). In particular, if X is irreducible
, With the preceding notation and assumptions, ?(X s ) is open in X//G, we have X s = ? ?1 ?(X s ) (in particular, X s is an open G-stable subset of X), and the restriction ? s : X s ?? ?(X s ) is a geometric quotient
he proved that all the orbits are closed and have the same dimension. In view of that and the previous result, if our group j k G is reductive, then conjecture 1 holds. However, the group j k G is not reductive. In fact, we can easily check that we have: 1. If deg xn P i = d ? 2, then PROJ(P 1 ,
P r ) contains all non constant PSRC j (P i , P k ) for j = 0, If 1 ? d = min deg xn P i , deg xn P k , then PROJ(P 1 ,
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