Algebraic Geometry by Andreas Gathmann

By Andreas Gathmann

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Xn ] we set n V (I) := {x ∈ P : f (x) = 0 for all homogeneous f ∈ I} ⊂ Pn . Obviously, if I is the ideal generated by a set S of homogeneous polynomials then V (I) = V (S). (c) If X ⊂ Pn is any subset we define its ideal to be I(X) := f ∈ K[x0 , . . , xn ] homogeneous : f (x) = 0 for all x ∈ X K[x0 , . . , xn ]. 10 we will denote them by Vp (S) and I p (X), and the affine ones by Va (S) and Ia (X), respectively. 12. (a) As in the affine case, the empty set 0/ = Vp (1) and the whole space Pn = Vp (0) are projective varieties.

Fr ) ⊂ Pn is a projective variety. Projective varieties that are of this form are called linear subspaces of Pn . 13. Let a ∈ Pn be a point. Show that the one-point set {a} is a projective variety, and compute explicit generators for the ideal I p ({a}) K[x0 , . . , xn ]. 14. Let f = x12 − x22 − x02 ∈ C[x0 , x1 , x2 ]. The real part of the affine zero locus Va ( f ) ⊂ A3 of this homogeneous polynomial is the 2-dimensional cone shown in the picture below on the left. 3 already that we should rather think of P2 as the affine plane A2 (embedded in A3 at x0 = 1) together with some points at infinity.

Then PnC is compact: let S = {(x0 , . . , xn ) ∈ Cn+1 : |x0 |2 + · · · + |xn |2 = 1} be the unit sphere in Cn+1 . This is a compact space as it is closed and bounded. Moreover, as every point in Pn can be represented by a unit vector in S, the restricted map π|S : S → Pn is surjective. Hence Pn is compact as a continuous image of a compact set. 5 (Homogeneous polynomials). In complete analogy to affine varieties, we now want to define projective varieties to be subsets of Pn that can be given as the zero locus of some polynomials in the homogeneous coordinates.

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