k is an algebraically closed field.
n dimensional affine space
A^{n} over k is
the set k^{n}. The components of a vector are its coordinates.
A^{1} and A^{2} are called the affine line and plane
respectively.
n dimensional projective space P^{n} over k is
the set of lines through the origin in k^{n+1}. P^{1} and P^{2} are called the projective line and plane
respectively. The coordinates of a vector in k^{n+1}{0} are
called the homogeneous coordinates of line
spanned by it.
There is (noncanonical) decomposition of P^{n} as a union
of A^{n} and P^{n1} where the
P^{n1} corresponds to the set where the last coordinate is 0.
When n=1, we can also see
this as follows: every line in k^{2} is determined by it's slope which
is either finite (in A^{1}) or infinity.
Given a set S of polynomials (respectively homogeneous polynomials) in k[x_{1},...x_{n}]
(respectively k[x_{0},...x_{n}]), let V(S) (respectively V_{P}(S)) be the
set of zeros of these
polynomials in A^{n} (respectively P^{n}).
An algebraic subset of ndimensional
affine (respectively projective) space
is a set of the form V(S) (V_{P}(S)). These are also called closed
sets since they are precisely closed sets for the Zariski topology.
When X is a subset of A^{n}, let I(X) be the set of polynomials in
k[x_{1},...x_{n}] 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
A^{n} and radical ideals in k[x_{1},... x_{n}]. 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 A^{n} 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 A^{n} is an open subset of its
closure in P^{n}.
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 A^{n}, then its coordinate ring A(X) = k[x_{1},...x_{n}]/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 kalgebras
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.
The category of quasiprojective varieties has products which refines the set
theoretic product. The product of A^{n} and A^{m} is isomorphic
to A^{n+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
P^{n}xP^{m} >
P^{nm+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[x_{1}...x_{n}] defines a hypersurface
V(f) in A^{n}. 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.
Two varieties are birational if
they have isomorphic function fields. Equivalently, two varieties are
birational if they have isomorphic open sets. Such and isomorphism
is called a birational equivalence.
