Introduction To Algebraic Geometry

Yuriy Drozd Intriduction to Algebraic Geometry Kaiserslautern 1998/99 CHAPTER 1 Affine Varieties 1.1. Ideals and varieties. Hilbert’s Basis Theorem Let K be an algebraically closed field. We denote by AnK (or by A if K is fixed) the n-dimensional affine space over K , i.e. the set of all n-tuples a = (a1 , a2 , . . . , an ) with entries from K . A subset X ⊆ AnK is called an affine algebraic variety if it coincides with the set of common zeros of a set of polynomials S = { F1 , F2 , . . . , Fm } ⊆ K[ x1 , . . . , xn ] . We denote this set by V (S) or V (F1 , F2 , . . . , Fm ) . We often omit the word “algebraic” and simply say “affine variety,” especially as we almost never deal with other sorts of varieties. If F is a subfield of K , one denotes by X(F) the set of all points of the variety X whose coordinates belong to F . If S consists of a unique polynomial F 6= 0 , the variety V (S) = V (F ) is called a hypersurface in An (a plane curve if n = 2 ; a space surface if n = 3 ). n Exercises 1.1.1. (1) Prove that the following subsets in An are affine algebraic varieties: (a) An ; (b) ∅ ; (c) { a } for every point a ∈ An . (d) (tk , tl ) | t ∈ K ⊂ A2 , where k, l are fixed integers. (2) Suppose that F = F1k1 . . . Fsks , where Fi are irreducible polynomials. Put X = V (F ) , Xi = V (Fi ) . Show that X = Ss i=1 Xi . (3) Let K = C be the field of complex numbers, R be the field of real numbers. Outline the sets of points X(R) for the plane curves X = V (F ) , where F are the following polynomials: (a) x2 − y 2 ; (b) y 2 − x3 (“cuspidal cubic”); (c) y 2 − x3 − x2 ; (“nodal cubic”); (d) y 2 − x3 − x (“smooth cubic”). (We write, as usually, (x, y) instead of (x1 , x2 ) , just as in the following exercise we write (x, y, z) instead of (x1 , x2 , x3 ) .) (4) Outline the sets of points X(R) for the space surfaces X = V (F ) , where F are the following