By Bhama Srinivasan (auth.)

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N. • The quotiented scaling action is encoded in the way the coordinates scale (all equally for Pn ), so this is combinatorial data. 1. Weighted Projective Spaces. Weighted projective spaces are deﬁned via diﬀerent torus actions. Consider the C∗ action on C4 deﬁned by λ : (X1 , X2 , X3 , X4 ) → (λw1 X1 , λw2 X2 , λw3 X3 , λw4 X4 ) (diﬀerent combinatorial data). We deﬁne P3(w1 ,w2 ,w3 ,w4 ) = C4 \ {0} /C∗ . Suppose w1 = 1. Then choose λ = 1 such that λw1 = 1. 1 Since this singularity appears in codimension 3, a subvariety of codimension 1 will generically not intersect it — so it may not cause any problems.

2. As an example of how a sheaf diﬀers from a vector bundle, consider Pn and the sheaf OÈn , the sheaf of holomorphic functions. This sheaf is also the sheaf of holomorphic sections of the trivial bundle, and the stalk over any point is the additive group of germs of holomorphic functions at that point. Now consider a subvariety V ⊂ Pn . We can consider OV , a sheaf over V, or we can consider a sheaf over Pn with support only along V. As a sheaf over Pn , OV can be deﬁned as holomorphic functions modulo holomorphic functions vanishing along V.

The coeﬃcients (1, 1, 1) in this relation encode the scaling action under λ ∈ C∗ : zi → λ1 zi . Note that we have introduced a coordinate for each vector. Note that the triple of vectors v1 , v2 , v3 are not all contained in a single cone, though any two of them are (there are three cones in the picture, the white areas). This encodes the data of the set U = {z1 = z2 = z3 = 0}. When we take C3 \ U, the scaling action has no ﬁxed points, and we can safely quotient by C∗ . The resulting smooth variety is, of course, P2 .