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Affine Variety

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

{(x,y,z):x^2+y^2-z^2=0}
(1)

is the cone, and

{(x,y,z):x^2+y^2-z^2=0,ax+by+cz=0}
(2)

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, the ideal of all polynomials vanishing on the cone is I(C)=(x^2+y^2-z^2) in the polynomial ring C[x,y,z]. The ideal of polynomials vanishing on the conic section is the ideal radical of the ideal generated by x^2+y^2-z^2 and ax+by+cz. For example, when the plane is x-z=0, this vanishing ideal is (x-z,y), whereas the ideal generated by the two defining equations is (x-z,y^2).

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.,

V={x=(x_1,...,x_n):p_1(x)=...=p_k(x)=0}
(3)

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. If the defining prime ideal has ideal height k, then V has dimension n-k. The number of listed generators alone does not determine the ideal height or the dimension. In either case, V may have singular points such as the origin in the cone.

A variety of dimension 1 is an algebraic curve, and one of dimension 2 is an algebraic surface. In complex affine space, the nonsingular part of an algebraic curve is a Riemann surface.

AffineVarieties

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}]
}}]

See also

Abhyankar-Sathaye Conjecture, Affine Scheme, Algebraic Set, Algebraic Variety, Category Theory, Commutative Algebra, Conic Section, Gröbner Basis, Lipman-Zariski Conjecture, Scheme, Zariski Cancellation Problem, Zariski Topology

Portions of 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.

Referenced on Wolfram|Alpha

Affine Variety

Cite this as:

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

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