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Varieties: Where Polynomials Become Shapes

An affine variety is just the solution set of polynomial equations — but read geometrically. We define affine space, the vanishing set, and our first examples: a circle, a parabola, a pair of lines, and the twisted cubic.

Affine space and the vanishing set

Fix a field k. Affine n-space over k, written A^n, is just the set k^n of n-tuples (a_1, …, a_n) — but we deliberately strip away the vector-space structure. There is no preferred origin, no addition we care about; A^n is a stage on which we will draw shapes cut out by polynomials. See affine space for the careful statement.

Given a set S of polynomials in the polynomial ring k[x_1, …, x_n], its vanishing set is V(S) = { p in A^n : f(p) = 0 for every f in S }. An affine variety is a set of this form. The slogan: a variety is the common zero locus of a family of polynomials. Read vanishing set and affine variety together — they are two halves of the same idea.

A gallery of first examples

Let us actually look at some. In A^2 the single polynomial x^2 + y^2 − 1 cuts out a circle; y − x^2 cuts out a parabola; xy cuts out the union of the two coordinate axes (a point kills xy iff x = 0 or y = 0). Already we see varieties that are 'one piece' and varieties that visibly split into pieces — the seed of irreducibility, which we study in guide 3.

Work in A^3 over an algebraically closed field k.
The TWISTED CUBIC is the image of t |-> (t, t^2, t^3).

As a variety it is V(I) for the ideal
    I = ( y - x^2,  z - x^3 ).

Check a point lies on it:  if y = x^2 and z = x^3, set t = x;
then (x, y, z) = (t, t^2, t^3).  Conversely every such triple
kills both generators.  So the curve = V(y - x^2, z - x^3).

Note: ONE equation in A^3 generally gives a SURFACE (2-dim);
to pin down a CURVE (1-dim) in A^3 we needed TWO equations.
Each independent equation tends to drop dimension by 1.

Warning that pays off later:  the ideal of this curve is
NOT just (y - x^2, z - x^3) plus the obvious; e.g. the relation
    x*z - y^2 = x*x^3 - (x^2)^2 = 0
also vanishes on it.  Generators of an ideal are subtle.
The twisted cubic: a curve in A^3 needs two equations, and its ideal hides extra relations.

Properties of V — the first half of the dictionary

The operation V reverses inclusions and behaves well with the lattice of ideals. These rules look bureaucratic, but they are precisely what will let V become a topology in guide 2.

  1. Order-reversing: if I ⊆ J then V(I) ⊇ V(J). More equations means fewer (or equal) solutions.
  2. Unions: V(I) ∪ V(J) = V(I ∩ J) = V(IJ). A point on either piece kills the product; conversely IJ ⊆ I ∩ J forces the reverse inclusion.
  3. Intersections: V(Σ I_α) = ∩ V(I_α), for ANY family, even infinite. Imposing all the equations at once is the same as intersecting all the solution sets.
  4. Extremes: V(0) = A^n (everything) and V(1) = ∅ (the unit ideal cuts out nothing — no point can make 1 vanish).