Affine Variety -- from Wolfram MathWorld

An affine variety is an algebraic variety contained in affine space. For example,

${(x,y,z):x^2+y^2-z^2=0}$

(1)

is the cone, and

${(x,y,z):x^2+y^2-z^2=0,ax+by+cz=0}$

(2)

is a conic section, which is a subvariety of the cone. The cone can be written to indicate that it is the variety corresponding to . Naturally, many other polynomials vanish on , in fact all polynomials in $I(C)={x^2+y^2-z^2}$ . The set is an ideal in the polynomial ring . Note also, that the ideal of polynomials vanishing on the conic section is the ideal generated by and .

A morphism between two affine varieties is given by polynomial coordinate functions. For example, the map is a morphism from to . Two affine varieties are isomorphic if there is a morphism which has an inverse morphism. For example, the affine variety is isomorphic to the cone via the coordinate change .

Many polynomials may be factored, for instance , and then . Consequently, only irreducible polynomials, and more generally only prime ideals are used in the definition of a variety. An affine variety is the set of common zeros of a collection of polynomials , ..., , i.e.,

$V={x=(x_1,...,x_n):p_1(x)=...=p_k(x)=0}$

(3)

as long as the ideal is a prime ideal. More classically, an affine variety is defined by any set of polynomials, i.e., what is now called an algebraic set. Most points in will have dimension , but may have singular points like the origin in the cone.

When is one-dimensional generically (at almost all points), which typically occurs when , then is called a curve. When is two-dimensional, it is called a surface. In the case of CW-complex affine space, a curve is a Riemann surface, possibly with some singularities.

The Wolfram Language function ContourPlot will graph affine varieties in the real affine plane. For example, the following graphs a hyperbola and a circle.

GraphicsGrid[{{
 ContourPlot[x^2 - y^2 == 1, {x, -2, 2}, {y, -2, 2}],
 ContourPlot[x^2 + y^2 == 1, {x, -2, 2}, {y, -2, 2}]
}}]

Affine Variety

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