Algebraic variety

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Algebraic varieties
are the central objects of study in algebraic geometry, a sub-field of mathematics.

Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers.

Modern definitions generalize this concept in several different ways,
while attempting to preserve the geometric intuition behind the original definition.[1]:58

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Many algebraic varieties are manifolds, but
an algebraic variety may have singular points while a manifold cannot.

Algebraic varieties can be characterized by their dimension.
Algebraic varieties of dimension one are called algebraic curves and
algebraic varieties of dimension two are called algebraic surfaces.

In the context of modern scheme theory,
an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field
whose structure morphism is separated and of finite type.

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An affine variety over an algebraically closed field is conceptually the easiest type of
variety to define.

In algebraic geometry, an affine variety, or affine algebraic variety,
over an algebraically closed field k is the zero-locus in the affine space k^n of
some finite family of polynomials of n variables with coefficients in k that
generate a prime ideal.

If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set.