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Proof. 2, we proceed by induction on d := dima. For d = 0 , there is nothing to prove since then a = 0,so X , = 0,= V, = T because of o" = MR. 9, T-objects like orbits, invariant irreducible closed subvarieties, and affine open invariant subvarieties in X u correspond to the respective T,-objects in 2, via Y ++ Y n 2, and vice versa. For convenience, we fix a closed equivariant embedding X ct C‘ given by the character functions xi = xp’ E Sx corresponding to a system of non-zero generators p l , .

P‘ of E also generate the corresponding cone y, the converse need not be true, even if the generating vectors of the cone are primitive; cf. 2 (2). 4. For the semigroup E, := y n M cut out by a strongly convex cone y, there is a canonical minimal system of generators, sometimes called a Hilbert basis: It is the set E \ (E + 8 ) of indecomposable elements in E , with E := E \ (0). In the two-dimensional case, it consists of those primitive lattice vectors p a which lie on the boundary of the (unbounded) “polyhedron” K := conv(8).

N} as above) are cyclic quotients Cn/Gj. There is an isomorphism P(a) z Pn/G(a) with G(a) := Cai acting coordinatewise on P,, so in particular, P(1,. . ,1) equals P,. Moreover, the description P, E (C"+')/D given above generalizes t o the weighted projective space if one replaces the diagonal 1-subtorus D c (C*)n+l with D(a) := {t" = ( P o , . , t a n ); t E C*}. 5 ny=o Secondly, there is a similar equivalence of suitable categories (cf. 7); see (3) below. 6. (1) For each cone c E A, there exists an associated orbit 0, := T*A,(O), where v E a ' .