oo We say that Xi is directly less than X} and write Xi -

E. as an S-scheme {e} = S). Hence in the case X = G x Y, the stack {XIG} can be considered as the quotient variety XI G = (G x y) I G = Y. e. also in this case, the stack {XIG} is isomorphic to Y. For stacks and their morphisms one may generalize many notions already known in the theory of schemes. First, we say that a morphism ¢ : :Fi -+ F2 of stacks is representable, if for every S-morphism 1/1 : X -+ F 2 , where X is an S-scheme, the stack X X:F2 FI is an S-space. Then the diagonal morphism 8 : F -+ F x F is representable if and only if, for every pair of morphisms of S-schemes Vi -+ F, i = 1,2, the fibre product VI X:F V2 is an S-space.

Then we have a commutative diagram: X ;/~ K*--------------·· r* where J-Lo is a moment map determined by X with the action of T together with (J) and the horizontal map K* --* r* is induced by inclusion r C K. Next, we have the following: I. LO) . gEG Now, let X be a smooth complex projective variety with an algebraic action of a complex algebraic group G, then X and the action can be considered as complex algebraic. ) C X of semistable points defined by these data: embedding and lifting (equivalently linearization of £.