k is an algebraically closed field.

n dimensional affine space An over k is the set kn. The components of a vector are its coordinates. A1 and A2 are called the affine line and plane respectively.

n dimensional projective space Pn over k is the set of lines through the origin in kn+1. P1 and P2 are called the projective line and plane respectively. The coordinates of a vector in kn+1-{0} are called the homogeneous coordinates of line spanned by it. There is (noncanonical) decomposition of Pn as a union of An and Pn-1 where the Pn-1 corresponds to the set where the last coordinate is 0. When n=1, we can also see this as follows: every line in k2 is determined by it's slope which is either finite (in A1) or infinity.

Given a set S of polynomials (respectively homogeneous polynomials) in k[x1,...xn] (respectively k[x0,...xn]), let V(S) (respectively VP(S)) be the set of zeros of these polynomials in An (respectively Pn). An algebraic subset of n-dimensional affine (respectively projective) space is a set of the form V(S) (VP(S)). These are also called closed sets since they are precisely closed sets for the Zariski topology.

When X is a subset of An, let I(X) be the set of polynomials in k[x1,...xn] which vanish on X. I(X) is always a radical ideal. The Hilbert Nullstellensatz states that I and V are inverse operations yielding a bijection between the collection of algebraic subsets of An and radical ideals in k[x1,... xn]. As a corollary, we have the weak Nullstellensatz V(I) is empty if and only if I = (1).

The set of complements of algebraic sets in affine or projective space forms a topology called the Zariski topology. Any subset gets an induced topology which goes by the same name. If k is a topological field such as C, then affine and projective space carries a second topology called the usual topology which is finer than the Zariski topology.

An affine or projective algebraic set is called a variety if it is irreducible in its Zariski topology i.e. if it cannot be written as the union of two proper closed sets. An algebraic subset X of An is a variety if and only if I(X) is prime.

A quasiprojective variety is an open subset of a projective variety. Projective varieties are clearly quasiprojective. Affine varieties are also quasiprojective, X in An is an open subset of its closure in Pn.

An algebraic map or regular map or morphism of quasiprojective varieties is a map of whose graph is closed. The collection of quasiprojective varieties and morphisms forms a category. An isomorphism of quasiprojective is an isomorphism in this category; more concretely it is a bijection for which it and its inverse are both morphisms.

If X is affine variety in An, then its coordinate ring A(X) = k[x1,...xn]/I(X). The functor X → A(X) is an antiequivalence of the full subcategory of affine varieties and the category of affine domains over k (finitely generated k-algebras which are integral domains). In particular, a morphism of affine varieties X → Y is an isomorphism if and only if the induced homomorphism A(Y) → A(X) is an isomorphism of algebras.

In classical geometry, the only ideals in k[x1,...xn] of interest are ideals I(X) associated to algebraic sets. These are precisely the radical ideals. In modern algebraic geometry, there is a more general object called a scheme . There is a one to one correspondence between the set of all ideals in k[x1,...xn] and closed subschemes of An. The coordinate rings of such things can have nilpotents.

The category of quasiprojective varieties has products which refines the set theoretic product. The product of An and Am is isomorphic to An+m. The product of two affine varieties is affine, and its coordinate ring is the the tensor product of the coordinate rings of the factors.

The product of projective varieties is projective thanks to the Segre embedding PnxPmPnm+n+m. The map sends a pair to the product of homogeneous coordinates of the pair.

An algebraic group is a group G in the category of quasiprojective varieties i.e. G is simulateneously a group and variety and the group multiplication G x G → G and inversion G → G are morphisms.

A homogenous space is a variety X such that there is an algebraic group G and a transitive action on X for which GxX → X is a morphism.

A nonconstant polynomial f in k[x1...xn] defines a hypersurface V(f) in An. A point a of V(f) is called a singular point if all the partial derivatives of f vanish at a. The definition can be extended to all varieties and it can be made more intrinsic. In particular an isomorphism takes a singular point to a singular point. When k=C, a nonsingular variety (i.e. a variety whose points are all nonsingular) is a complex manifold.

The function field of an affine variety is the quotient field of its coordinate ring. An element of the function field, called a rational function, can be represented by a regular function on a nonempty open set. This leads to a defintion of function field for quasiprojective varieties.

The dimension of an algebraic variety X is the length n of longest chain X0 ⊂ X0 ⊂ ... X0=X of closed sets. If X is affine, then this coincides with the Krull dimension of A(X). The dimension also coincides with the transcendence degree of the function field. The dimension of An and Pn are both n. If X is a hyypersurface in An or Pn, its dimension is n-1.

Two varieties are birational if they have isomorphic function fields. Equivalently, two varieties are birational if they have isomorphic open sets. Such an isomorphism is called a birational equivalence. Birational varieties have the same dimension.