By Conforto F.

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**Example text**

Z through some point x E X \ {0} = Xa,. We may thus replace x with the orbit base point x,,,. By hypothesis, the unicity of 7-0 is valid in every affine open subvariety X,, that includes 0, so it is valid in X u . For the asserted correspondence between faces and invariant irreducible closed subvarieties, we consider such an A L) X , with A := A \ F # 8. Then A = UTEa,(A n X,) implies the equality dim(A n X,) = dimA for some face T E 8u. Again by induction hypothesis, there is a face TO 5 T with A n X, = V(TO) = TO), the closure being taken in X,; moreover, TO) is open in AnX, and hence, has the same dimension.

For any two cones u,u’ E A, the intersection X u n Xu, is a T-invariant affine open subspace of X, and thus X u n Xu) = X, with a cone r E A. Since X is separated, a one-parameter subgroup X E Y(T) has at most one limit 41 X(0) E X . 8 (2), the following holds: I - n N = {V E N ; x,(o) E xunXul} Xu}n {V =(anN)n(dnN). = {V E N ; X,(o) E E N ; X,(O) E Xul} This readily implies that I- = a n a’. 3. An (N-lattice) fan in Nw is a finite non-empty set A of (strongly convex) N-cones satisfying (1) r 5 u E A j T E A ; (2) a, a’ E A ----7- a n a’ 5 0,a’; in particular, u n a’ E A.

8, we know that t o each a6ne toric variety, say U , corresponds a unique N-cone u = uu such that U = X u . To the general toric variety X, we may thus associate the following collection of N-cones: A := A(X) := {u = m~ E Ob(6CN) ; U QI X} , where U runs through the affine open toric subvarieties of X. For any two cones u,u’ E A, the intersection X u n Xu, is a T-invariant affine open subspace of X, and thus X u n Xu) = X, with a cone r E A. Since X is separated, a one-parameter subgroup X E Y(T) has at most one limit 41 X(0) E X .