Subvariety

A subset of an algebraic variety which is itself a variety. Every variety is a subvariety of itself; other subvarieties are called proper subvarieties.

A sphere of the three-dimensional Euclidean space is an algebraic variety since it is defined by a polynomial equation. For example,

(1)

defines the sphere of radius 1 centered at the origin. Its intersection with the -plane is a circle given by the system of polynomial equations:

			(2)
			(3)

Hence the circle is itself an algebraic variety, and a subvariety of the sphere, and of the plane as well.

Whenever some new independent equations are added to the equations defining a certain variety, the resulting variety will be smaller, since its points will be subject to more conditions than before. In the language of ring theory, this means that, while the sphere is the zero set of all polynomials of the ideal of , every subvariety of it will be defined by a larger ideal; this is the case for , which is the defining ideal of the circle.

In general, given a field , if is the affine variety of defined by the ideal of , and is an ideal containing , then (if it is nonempty) is a subvariety of . If is an algebraically closed field, it follows from Hilbert's Nullstellensatz that iff ; in this case, is a proper subvariety of iff . The same applies to projective algebraic varieties and homogeneous ideals.