MATH 8253 ALGEBRAIC GEOMETRY WEEK 11 CIHAN_ BAHRAN 3.1.8. Let X ! S be a surjective morphism. Show that X ×S Y ! Y is surjective for any S-scheme Y . In other words, surjective morphisms are stable under base change. (Use Proposition 1.16) n 3.1.11. Let k be a field and z 2 Pk . We choose a homogenous coordinate system such that z = (1; 0;:::; 0). n−1 (a) Show that there exists a morphism p : Pkn r fzg ! Pk such that over k, where k is the algebraic closure of k, we have pk(a0; a1; : : : ; an) = (a1; : : : ; an) for every point of n(k). Such a morphism is called a projection with cen- Pk ter z. n (b) Let X be a closed subset of Pk not containing z. Show that X cannot −1 n−1 contain p (y) for any y 2 Pk . 3.3.1. Let f : X ! Y be a morphism of schemes. We suppose that there exist open S −1 −1 subsets Yi of Y such that X = i f (Yi), and that the restrictions f : f (Yi) ! Yi are closed immersions. Show that if f(X) is closed in Y , then f is a closed immersion. 3.3.2. Let X be a scheme. Show that the following properties are equivalent: (i) X is separated; (ii) X is separated over an affine scheme; (iii) for any scheme Y , every morphism of schemes X ! Y is separated. 3.3.3. Let f : X ! Y be a surjective closed morphism between two Noetherian schemes (or, more generally, topological spaces). Show that we have the inequality dim X ≥ dim Y (reduce to the case when X is irreducible and use induction on dim X). Compare with Exercises 2.5.8 and 4.3.3. n n 3.3.4. Let k be a field. Let Z = Ak or Pk . Let us consider two closed sub varieties X; Y of Z, pure of respective dimensions q; r. (a) Show that the irreducible components X ×k Y are of dimension q + r. (b) Show that dim(X [ Y ) ≥ q + r − n if X \ Y 6= ? (identify X \ Y with the subset 4(Z) \ (X ×k Y ) of Z ×k Z and use Corollary 2.5.26). n (c) If Z = Pk and q + r ≥ n, show that X \ Y 6= ? (if X = V+(I), Y = V+(J), n+1 consider V (I) \ V (J) in Ak and use Corollary 2.5.21). (d) Show that every projective plane curve X (i.e., X is a closed sub variety of 2 Pk pure of dimension 1) is connected. 1 MATH 8253 ALGEBRAIC GEOMETRY WEEK 11 2 3.3.6. Let f : X ! Y be a separated morphism. Show that any section of f is a closed immersion (apply Proposition 3.9(f) to X ×X Y ! X ×Y Y ) 3.3.7. Let X be a Y -scheme. Show that X is separated over Y if and only if for 0 0 0 every Y -scheme Y , the sections of X ×Y Y ! Y are closed immersions (hint: 4X=Y is a section of the second projection X ×Y X ! X). 3.3.8. Let f : X ! Y be a morphism with X affine and Y separated. Show that f −1(V ) is affine for every affine open subset V of Y (use Proposition 3.9(f)). 3.3.10. (Graph of a morphism) Let f : X ! Y be a morphism of S-schemes. We define the graph Γf of f as being the image of (idX ; f): X ! X ×S Y . (a) Show that (idX ; f) is a section of the projection morphism X ×S Y ! X. Deduce from this that Γf is a closed subset of X ×S Y if Y is separated over S. (b) Let us suppose X is reduced and let us endow Γf with the structure if a reduced closed subscheme. Show that if Y is separated over S, then the projection X ×S Y ! X induces an isomorphism from Γf onto X. 2 3.3.21. Let k be a field. Show that two closed subsets of dimension 1 in Pk always have a non-empty intersection (use Exercise 3.4). Deduce from this that 1 1 2 Pk ×k Pk Pk. 3.3.26. 3.3.27..