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# 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 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.

```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 Scheme, Algebraic Set, Algebraic Variety, Category Theory, Commutative Algebra, Conic Section, Gröbner Basis, Scheme, Zariski Topology

This entry contributed by Todd Rowland

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## References

Bump, D. Algebraic Geometry. Singapore: World Scientific, pp. 1-6, 1998.Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms. New York: Springer-Verlag, pp. 5-29, 1997.Hartshorne, R. Algebraic Geometry. New York: Springer-Verlag, 1977.

Affine Variety

## Cite this as:

Rowland, Todd. "Affine Variety." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/AffineVariety.html