Chapter 1 Affine and Projective Varieties In this chapter we define an affine variety as the set of points of the affine space An over the field C which are common zeros of a finite set of poly- nomials in C[X1,...,Xn]. An important result is Hilbert’s Nullstellensatz which establishes a bijective correspondence between affine varieties of An and radical ideals of the polynomial ring C[X1,...,Xn]. We define analo- gously projective varieties of the projective space Pn as the set of common zeros of a finite set of homogeneous polynomials, and we state a projective Nullstellensatz. 1.1. Affine varieties Let C[X1,X2,...,Xn] denote the ring of polynomials in the variables X1,X2, . . . , Xn over C. The set Cn = C × · · · × C will be called affine n-space and denoted by An C or just An. We define an affine variety as the set of common zeros in An C of a finite collection of polynomials in C[X1,...,Xn]. To each ideal I of C[X1,...,Xn] we associate the set V(I) of its common zeros in An C . By Hilbert’s basis theorem, the C-algebra C[X1,...,Xn] is Noetherian hence each ideal of C[X1,...,Xn] has a finite set of generators. Therefore the set V(I) is an affine variety. To each subset S An C we associate the collection I(S) of all polynomials vanishing on S. It is clear that I(S) is an ideal and that we have inclusions S V(I(S)), I I(V(I)), which are not equalities in general. Example 1.1.1. If f C[X1,X2,...,Xn] \ C, the affine variety V(f) is called a hypersurface of An C . 3
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