By Ulrich Gortz, Torsten Wedhorn

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R. Deduce that for any finite subset X ⊂ Pn (k) there exists a hyperplane Z of Pn (k) such that X ∩ Z = ∅. 27. Let Y be a quasi-projective variety. Show that every finite subset of Y is contained in an open affine subvariety of Y . 26. 28♦. Let n ≥ 0 be an integer. 15) and let L0 ⊂ k n+1 be a fixed one-dimensional subvector space. Show that the map G → Pn (k), g → g(L0 ) is a surjective morphism of prevarieties. 29. Let X be an affine variety. (a) Show that any morphism Pn (k) → X is constant. 41 that this is the case if Z is a projective variety).

23) that two different lines in the projective plane always meet in one point. 1) Pn (k) = {lines through the origin in k n+1 } = (k n+1 \ {0})/k × . Here a line through the origin is per definition a 1-dimensional k-subspace and we denote by (k n+1 \ {0})/k × the set of equivalence classes in k n+1 \ {0} with respect to the equivalence relation (x0 , . . xn ) ∼ (x0 , . . , xn ) ⇔ ∃λ ∈ k × : ∀i : xi = λxi . 1) is given by attaching to the equivalence class of (x0 , . . , xn ) the 1-dimensional subspace generated by this vector.

21) Projective varieties. 62. A prevariety is called a projective variety if it is isomorphic to a closed subprevariety of a projective space Pn (k). As in the affine case, we speak of projective varieties rather than prevarieties. Similarly, we will talk about subvarieties of projective space, instead of subprevarieties. For an explanation why this is legitimate, we refer to Chapter 9. For x = (x0 : · · · : xn ) ∈ Pn (k) and f ∈ k[X0 , . . , Xn ] the value f (x0 , . . , xn ) obviously depends on the choice of the representative of x and we cannot consider f as a function on Pn (k).

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