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 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.,

(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.