nLab
algebraic variety

Idea

Classically, an algebraic variety is thought of as a geometric locus of zeros of a set of polynomial equations in finitely many variables.

Historically, there were several formalisms of various schools including the Italian school of algebraic geometry in the early 20th century (Veronese, Castelnuovo, Severi, …), the American school between the two wars (Oscar Zariski?), Andre Weil?, the abstract varieties of Jean-Pierre Serre and finally the scheme language of the Grothendieck school. One should note that in the case of (esp. projective) varieties over complex numbers there is an additional possibility to work using complex-analytic tools and complex topology.

Definition

Given an algebraically closed field $k$, an algebraic $k$-variety is an affine, quasiaffine, projective or quasiprojective $k$-variety. Affine $k$-varieties are maximal spectra (= sets of maximal ideals) of finitely generated noetherian (commutative unital) $k$-algebras without nilpotents with the Zariski topology; the algebra can be recovered as the coordinate ring of the variety; this correspondence is an equivalence of categories, if the morphisms are properly defined. Affine varietes can be embedded as closed subvarieties into an affine space (in the sense of algebraic geometry). As topological spaces affine varieties are noetherian. Projective $k$-varieties are obtained in a similar way from graded $k$-algebras. Quasiaffine $k$-varieties are Zariski-open subspaces of affine $k$-varieties; quasiprojective $k$-varieties are Zariski-open subspaces of projective $k$-varieties. In fact, by noticing that the affine $k$-space is Zariski open in a projective space of the same dimension, we see that the quasiprojective case includes all others. Morphisms between varieties are so-called regular map?s. Note that every kind of algebraic variety above may be interpreted as a quasiprojective variety.

Properties

There is an equivalence of categories between the category of integral schemes of finite type over $\mathrm{Spec}k$, where $k$ is an algebraically closed field, and the category of algebraic $k$-varieties.

Of course, the corresponding scheme and variety have different sets of points; the points in common are the closed points of the scheme. The remaining points are the generic points of subvarieties. Generic points were often used, without proper foundations, in other language, already in the works of the Italian school. Some modern algebraic geometers mean, by varieties, objects of certain slightly bigger categories of relative $S$-schemes (where $S$ is not necessarily $\mathrm{Spec}k$).

References

An amusing discussion on the differences between schemes and varieties can be found at Secret blogging seminar: algebraic geometry without prime ideals.