Algebraic variety

An algebraic variety is most directly realized as the common zero set of polynomials over a field, usually an algebraically closed field such as the complex numbers. In affine space, equations such as y−x²=0 define an affine variety; homogeneous equations instead define projective varieties inside projective space.¹ Different conventions vary on whether a variety must be irreducible, nonsingular, or defined over an algebraically closed field. Modern algebraic geometry commonly treats a variety as a reduced, separated scheme of finite type over a field, while classical texts often begin with point sets and polynomial ideals.²

The polynomial ideal records every equation vanishing on the set. Hilbert's Nullstellensatz connects such ideals with geometric zero sets and makes radical ideals central to the correspondence between algebra and geometry.³

The coordinate ring translates geometric questions about an affine variety into commutative algebra. Prime ideals correspond to irreducible subvarieties, maximal ideals describe ordinary points in the classical setting, and chains of prime ideals measure dimension.¹ A curve has dimension one, a surface has dimension two, and dimension generally behaves like the number of independent local parameters, although singular points can complicate this intuition. A morphism between varieties is given locally by regular functions; it is the algebraic counterpart of a structure-preserving map.

Important properties include irreducibility, smoothness, properness, and rationality. A smooth variety has no algebraic singularity, while a singular variety may have a cusp, node, or more complicated local defect. Projective varieties are especially useful because projective space supplies a form of algebraic compactness: maps and intersections often have better finiteness properties than their affine analogues.²

Classical examples range from lines and conics to elliptic curves, Grassmannians, and algebraic surfaces. A nonsingular plane cubic becomes an elliptic curve after choosing a rational point, and its points carry an abelian-group structure that links geometry with number theory. A Riemann surface can sometimes be represented by a smooth complex algebraic curve, making complex analysis and algebraic geometry meet in a particularly rich case.⁴

Varieties over finite fields support questions about point counts and led to the Weil conjectures, whose proof connected algebraic geometry with topology and étale cohomology. Arithmetic geometry studies varieties over number fields and finite fields, while moduli theory constructs varieties or related spaces whose points classify other geometric objects. These applications include Diophantine equations, coding theory, cryptography, and the study of parameter spaces.

The passage from varieties to schemes preserves the geometric intuition while allowing nilpotent elements, non-algebraically closed base fields, and families with changing fibers. A scheme can therefore retain information invisible in its ordinary set of points, including multiplicity and infinitesimal behavior.¹ This extension was developed through work associated with Alexander Grothendieck and became foundational for modern intersection theory and arithmetic geometry.

Not every geometric object described by polynomial equations is a variety under every convention. Reducible zero sets may contain several components, and nonreduced schemes may have the same points as a variety while carrying extra algebraic structure. Quotients and parameter spaces can also fail to be varieties in a straightforward sense, motivating algebraic spaces, stacks, and other generalizations. Tropical geometry provides a further, piecewise-linear shadow of varieties that preserves selected combinatorial and arithmetic information.

Affine variety

A variety realized as the common zero set of polynomials in affine space.

Projective variety

A variety defined by homogeneous polynomial equations in projective space.

Coordinate ring

The ring of polynomial functions on an affine variety, usually formed as a polynomial ring modulo its defining ideal.

Irreducible

A property meaning that the variety cannot be expressed as the union of two proper closed subvarieties.

Scheme

A generalization of a variety built from locally ringed spaces, allowing nilpotents and broader base fields.