By Andreas Gathmann

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10 √ tells us that I = (1) or I = (x0 , . . , xn ). In any case, this means that xiki ∈ I for some ki , so (x0 , . . , xn )k0 +···+kn ⊂ I. 6. There is a one-to-one inclusion-reversing correspondence between algebraic sets in Pn and homogeneous radical ideals in k[x0 , . . , xn ] not equal to (x0 , . . , xn ), given by the operations Z(·) and I(·). Proof. 5. 3. Projective varieties as ringed spaces. So far we have defined projective varieties as topological spaces. 1. So let X ⊂ Pn be a projective variety.

3. 1. g. the affine variety X = {(x, y) ; xy = 1} ⊂ A2 and the projection morphism f : X → A1 , (x, y) → x. The image of f is A1 \{0}, which is not closed in A1 . 13 why it is not closed: the “vertical point at infinity”, which would map to x = 0 ∈ A1 and make the image closed, is missing in the affine variety X. y X x f (X) = A1 \{0} To prove the above mentioned statement we start with a special case (from which the general one will follow easily). 2. e. if X ⊂ Pn × Pm is closed then so is π(X).

This is clear from the picture below as the hyperbola x1 x2 = 1 extends to infinity both along the x1 and the x2 axis. x2 x2 X1 X2 x1 x1 Note that the equations of X˜1 and X˜2 are exactly the same, up to a permutation of the coordinates. Even if we have not given projective varieties the structure of varieties yet, it should be obvious that X˜1 and X˜2 will be isomorphic varieties, with the isomorphism being given by exchanging x0 and x1 . Hence we see that the two distinct types of conics in A2 become the same in projective space: there is only one projective conic in P2 up to isomorphism.