By Andreas Gathmann

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3. Let S(X)(d) := { f (d) ; f ∈ S(X)} be the degree-d part of S(X). 8. We define the field of rational functions to be f ; f , g ∈ S(X)(d) and g = 0 . 2, the elements of K(X) give set-theoretic functions to the ground field k wherever the denominator is non-zero. Now as in the affine case set f OX,P := ∈ K(X) ; g(P) = 0 and OX (U) := OX,P g P∈U for P ∈ X and U ⊂ X open. It is easily seen that this is a sheaf of k-valued functions. 4. 9) one can show that the function field K(X) defined above agrees with the definition for general varieties.

Let F : X → P1 be the morphism defined by P → ( f (P) : g(P)); this is well-defined as f is non-zero on X by assumption. 7 the image F(X) is closed in P1 . Moreover, F(X) is irreducible as X is. Therefore, F(X) is either a point or all of P1 . But by assumption (0 : 1) ∈ / F(X), so F(X) must be a single point. But this is a contradiction, as F(P) = ( f (P) : g(P)) = (1 : 0) and F(Q) = ( f (Q) : g(Q)) = (1 : 0) by the choice of g. 3. 9. g. X = {x = 0} ⊂ A2 and f = x − 1, then X ∩ Z( f ) = 0/ although X is a line (and therefore contains more than one point).

X0 x0 denotes the homogenization of f with respect to x0 . Show that: (i) I h is a homogeneous ideal. (ii) Z(I h ) ⊂ Pn is the closure of Z(I) ⊂ An in Pn . We call Z(I h ) the projective closure of Z(I). (iii) Let I = ( f1 , . . , fk ). Show by an example that I h = ( f1h , . . , fkh ) in general. 4. In this exercise we will make the space of all lines in Pn into a projective variety. Fix n ≥ 1. We define a set-theoretic map ϕ : {lines in Pn } → PN with N = n+1 − 1 as follows. For every line L ⊂ Pn choose two distinct points P = 2 (a0 : · · · : an ) and Q = (b0 : · · · : bn ) on L and define ϕ(L) to be the point in PN whose homogeneous coordinates are the n+1 maximal minors of the matrix 2 a0 b0 ··· ··· an bn , in any fixed order.

### Algebraic Geometry by Andreas Gathmann

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