A variety is a polynomial zero set with geometric teeth
Classical algebraic varieties are solution sets of polynomial systems over the reals or complexes. Hilbert's Nullstellensatz then locks those shapes to ideals in polynomial rings, so geometric questions become ring-theoretic ones—and singular points are allowed where smooth manifolds forbid them.
Definitions vary slightly: some authors demand irreducibility in the Zariski topology and call reducible objects algebraic sets; others allow reducible varieties. Dimension one yields algebraic curves; dimension two, algebraic surfaces. Many varieties are differentiable manifolds away from singularities, but singularities are permitted. In scheme language a variety over a field is often an integral scheme that is separated and of finite type. Affine varieties over an algebraically closed field are the simplest: in affine n-space, the zero locus of a set of polynomials is an affine algebraic set, and an irreducible nonempty one is an affine variety, carrying the Zariski topology whose closed sets are algebraic sets.
Projective varieties use homogeneous polynomials on projective space, where only vanishing—not a raw value—is well-defined on homogeneous coordinates. Quasi-projective varieties are Zariski-open pieces of projective ones; every affine variety is quasi-projective in a standard chart. Classically, Hartshorne's early chapters treat varieties as quasi-projective, then later allow abstract varieties that look locally quasi-projective without a global projective embedding. Embedding-dependent definitions obscure intrinsic notions such as regular functions and force awkward moves like the Segre embedding before P¹ × P¹ counts as a variety.
André Weil tried an abstract definition via valuations; Claude Chevalley proposed schemes; Alexander Grothendieck's schemes won widest acceptance. In that language the classical objects are schemes that stay integral and separated, of finite type, and quasiprojective over a field that is algebraically closed. Nagata produced early examples of non-quasiprojective varieties, including a complete non-projective algebraic surface; complete but non-quasi-projective toric varieties are now routine. Hilbert's Nullstellensatz pairs closed subvarieties with prime ideals (or non-irrelevant homogeneous primes) of the coordinate ring. The fundamental theorem of algebra already hinted at the algebra–geometry link for one variable; Nullstellensatz is the several-variable backbone.
Source: Algebraic variety
