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By Andreas Gathmann

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Consider the map f : An → Pn , (x1 , . . , xn ) → (1 : x1 : · · · : xn ). As in the picture below on the left we can embed the affine space An in K n+1 at the height x0 = 1, and then think of f as mapping a point to the 1-dimensional linear subspace spanned by it. 6. 47 (1 : a1 : a2 ) x0 A2 Projective Varieties I: Topology 1 x0 1 A2 (a1 , a2 ) x(t) x2 b x1 b x1 The map f is obviously injective, with image U0 := {(x0 : · · · : xn ) : x0 = 0}. , , x0 x0 it sends a line through the origin to its intersection point with An embedded in K n+1 .

V there is nothing in the definition that guarantees us such an interpretation, and consequently there is no well-defined notion of composing these sections of the structure sheaves with the map f : X → Y . 2 (Sheaves = sheaves of K-valued functions). For every sheaf F on a topological space X we will assume from now on that the rings F (U) for open subsets U ⊂ X are subrings of the rings of all functions from U to K (with the usual pointwise addition and multiplication) containing all constant functions, and that the restriction maps are the ordinary restrictions of such functions.

Obviously, the only way to obtain the relations fX = πX ◦ f and fY = πY ◦ f is to take the map f : Z → X ×Y, z → ( fX (z), fY (z)). 7: as fX and fY must be given by regular functions in each coordinate, the same is then true for f . 11. 8. Hence the the coordinate ring A(X ×Y ) of the product is just the tensor product A(X) ⊗K A(Y ). 12. An affine conic is the zero locus in A2 of a single irreducible polynomial in K[x1 , x2 ] of degree 2. Show that every affine conic over a field of characteristic not equal to 2 is isomorphic to exactly one of the varieties X1 = V (x2 − x12 ) and X2 = V (x1 x2 − 1), with an isomorphism given by a linear coordinate transformation followed by a translation.

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