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, Projective Algebraic Variety, Scheme, Intrinsic Variety, Moduli Space Stack, 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.

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Affine Variety

Cite this as:

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

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