By Grauert G. (ed.)

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**Example text**

Fn). (2) The affine quadric cone Y corresponds to (T = cone(2fl- f2, f 2 ) (see Figure 9). 2 corresponds to 0 = cone (kfl-(k--l)f2, f 2 ) . (3) The Segre cone 2 corresponds to u = cone(f1, f 2 , f3, f1 + f 2 - f 3 ) . 4. To the face relation T 5 (T of N-cones corresponds an open embedding X , + X , of affine toric varieties. Proof. 5, the "face equality" some p E uv n M ) translates into rv = (T' Rep = u" + T = (T n p' + ray(-p). * .......... @ . *. Figure 9. The N-cones for C2 and Y turn implies that O(X,) = O(X,)[x-'], where x := x p .

The fan for the projective plane P2 Proof. We write N , in order to indicate the rank of the lattice under consideration; thus A is a fan in (N,)w. In N,+l, we exceptionally denote the standard lattice basis by (go,. . ,gn), and consider the regular cone a := cone(g0,. . ,g,). The homomorphism p : T,+l - T, , u := (uo,. . ,un) - - , (u1uO1,. . unu;l) induces the linear map d p : (N,+l)W n -Cfi,gi (Nn)w, go H H fz for i = 1 , . . , n , i=l which maps the cones in da onto the cones in A. Hence, p extends to a morphism cnfl\ { 0) = X& --$ XA .

More generally, the equivalence of categories stated for the affine case as in Expression (20) easily carries over to lattice fans and general toric varieties. ) (4) For fans A and A in Nw, we assume that each cone of A is included in some cone of A. The resulting morphism X A + X A is proper if and only if lA( = lAl; and in that case A is called a subdivision or refinement of A. - In the strong topology of complex varieties, “proper” means that the inverse image of a compact subset is again compact.