By Brian Osserman

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Many of our results so far can be reduced to the affine case, and thus extend to general prevarieties. 17. We illustrate a few routine generalizations in the following exercises. 19. Let X be a prevariety. (a) If U ⊆ X is open, then dim U = dim X. (b) If Z ⊆ X is closed and irreducible, then dim X = dim Z + codimX Z. We also have the following basic fact. 20. Let ϕ : X → Y be a morphism of varieties. Let Z ⊆ X be the closure of ϕ(X). Then Z is a variety, and dim Z dim X. 46 CHAPTER 6 Projective varieties We now move on to studying projective varieties, which we will treat as examples of the more general abstract varieties we have defined.

8. 4. We can see explicitly that the diagonal is not closed in this case: indeed, X × X has an atlas consisting of U1,1 , U1,2 , U2,1 , U2,2 where each Ui,j is a copy of A1k × A1k : the diagonal ∆ in X × X restricts to the diagonal in U1,1 and U2,2 , so is closed on these open subsets, but ∆|U1,2 and ∆|U2,1 are each equal to the complement of the origin in the diagonal of A1k × A1k , so are not closed. Thus, ∆ is not closed. Put differently, if P1 , P2 denote the two origins in X, then we see from the above atlas on X ×X that while the points (P1 , P1 ) and (P2 , P2 ) are in the diagonal, the points (P1 , P2 ) and (P2 , P1 ) are in the closure of the diagonal, but not in the diagonal.

If we tried to use x2 to compute the tangent space, we would think that the tangent space is all of k. We will see that the dimension of TP (Z) is always at least as large as the largest dimension of a component of Z containing P , and it will turn out that a point P of Z is nonsingular if and only if equality holds. However, we do not make this the definition, because it appears to depend on an imbedding in affine space. We will instead develop a definition which is visibly intrinsic. 2. Zariski cotangent spaces We now describe a notion of (non)singularity which is intrinsic, and which also generalizes well.