Affine Variety

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


is the cone, and


is a conic section, which is a subvariety of the cone. The cone can be written V(x^2+y^2-z^2) to indicate that it is the variety corresponding to x^2+y^2-z^2=0. Naturally, many other polynomials vanish on V(x^2+y^2-z^2), in fact all polynomials in I(C)={x^2+y^2-z^2}. The set I(C) is an ideal in the polynomial ring C[x,y,z]. Note also, that the ideal of polynomials vanishing on the conic section is the ideal generated by x^2+y^2-z^2 and ax+by+cz.

A morphism between two affine varieties is given by polynomial coordinate functions. For example, the map phi(x,y,z)=(x^2,y^2,z^2) is a morphism from X=V(x^2+y^2+z^2) to Y=V(x+y+z). Two affine varieties are isomorphic if there is a morphism which has an inverse morphism. For example, the affine variety V(x^2+y^2+z^2) is isomorphic to the cone V(x^2+y^2-z^2) via the coordinate change phi(x,y,z)=(x,y,iz).

Many polynomials f may be factored, for instance f=x^2+y^2=(x+iy)(x-iy), and then V(f)=V(x+iy) union V(x-iy). Consequently, only irreducible polynomials, and more generally only prime ideals p are used in the definition of a variety. An affine variety V is the set of common zeros of a collection of polynomials p_1, ..., p_k, i.e.,


as long as the ideal I=<p_1,...,p_k> 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 V will have dimension n-k, but V may have singular points like the origin in the cone.

When V is one-dimensional generically (at almost all points), which typically occurs when k=n-1, then V is called a curve. When V 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.

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

See also

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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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.

Referenced on Wolfram|Alpha

Affine Variety

Cite this as:

Rowland, Todd. "Affine Variety." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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