By Andreas Gathmann

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A prevariety is a ringed space X that has a finite open cover by affine varieties. Morphisms of prevarieties are simply morphisms as ringed spaces. 1, the elements of OX (U) for an open subset U ⊂ X will be called regular functions on U. 2. 1 is not part of the data needed to specify a prevariety — it is just required that such a cover exists. Any open subset of a prevariety that is an affine variety is called an affine open set. 3. Of course, any affine variety is a prevariety. 17. The basic way to construct new prevarieties is to glue them together from previously known patches.

Xn ]/I, where I is the kernel of g. If we assume that I is radical (which is the same as saying that R does not have any nilpotent elements except 0) then X = V (I) is an affine variety in An with coordinate ring A(X) ∼ = R. Note that this construction of X from R depends on the choice of generators of R, and so we can get different affine varieties that way. 8 implies that all these affine varieties will be isomorphic since they have isomorphic coordinate rings — they just differ in their embeddings in affine spaces.

We call it the prevariety obtained by gluing the Xi along the isomorphisms fi, j . 8. Show: (a) Every morphism f : A1 \{0} → P1 can be extended to a morphism A1 → P1 . (b) Not every morphism f : A2 \{0} → P1 can be extended to a morphism A2 → P1 . (c) Every morphism f : P1 → A1 is constant. 9. (a) Show that every isomorphism f : P1 → P1 is of the form f (x) = where x is an affine coordinate on A1 ⊂ P1 . ax+b cx+d for some a, b, c, d ∈ K, (b) Given three distinct points a1 , a2 , a3 ∈ P1 and three distinct points b1 , b2 , b3 ∈ P1 , show that there is a unique isomorphism f : P1 → P1 such that f (ai ) = bi for i = 1, 2, 3.

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Algebraic Geometry by Andreas Gathmann

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