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7. Affine Variety

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

(1)

is the cone, and

(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 . 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 areisomorphic 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.,

(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.

Mathematica's 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}] }}]
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