In this section, we want to sort out the various meanings of the word "point," as used by algebraic geometers. This word is used in at least three distinct ways. As a result, beginners in the subject often find themselves confused.

Let `k` be a field, and let `S` be
an algebraic variety over `k`. (An algebraic
variety is a scheme of finite type over a field. This
usage differs from that of other authors, who reserve the word variety
for an irreducible scheme of finite type. We want to work over
arbitrary fields, and the property of being irreducible is not stable
under the operation of extending the base field.) The most primitive
notion of a point is also the rarest usage among algebraic
geometers. We call any element of the underlying topological space of
`S` a point. When using the term this way, we often
make special mention of the closed points. This term
emphasizes the fact that the underlying topological space is not
Hausdorff, and many of its points are, in fact, not closed in
`S`.

Let's assume for the moment that the variety in question can be
embedded in a projective space `P ^{n}_{k}`. Then we
can give an elementary definition of a

If the field `k` is not algebraically closed, then
the `k`-rational points usually form a very small
subset of the collection of all closed points of `S`.
We can reinterpret the definition of a `k`-rational
point in a coordinate-free manner that will allow us to generalize the
above definition in several ways. Notice that to give a
`k`-rational point of `S` (in the
sense described above) is equivalent to specifying a
`k`-morphism `Spec(k)→ S`.
The first generalization, then, is to drop the assumption that
`S` can be embedded in projective space. Next,
suppose that `s∈ S` is a closed point. Then the
residue field `k(s) = O _{S,s}/m_{s}` is an extension
field of

We are now ready to define yet another kind of point on a
`k`-scheme `S`. There is no reason
to restrict the previous definition to finite extension fields of
`k`. If we let `k ^{-}` denote the
algebraic closure of

`
S(k ^{-}) = {α : Spec(k^{-}) → S} =
Hom_{k}(Spec(k^{-}), S).
`

`
S(T) = Hom _{k}(T, S).
`

`
° f : S(U) = Hom _{k}(U, S) → Hom_{k}(T, S) = S(T).
`

`
S(T) = Mor _{k}(T,S) = Hom_{k}(k, Γ(T, O_{T}))
`

`
S(T) = Mor _{k}(T,S) = Hom_{k}(k[x], Γ(T, O_{T})) = Γ(T,
O_{T}).
`

`
S(T) = Mor _{k}(T,S) = Hom_{k}(k[x, x^{-1}], Γ(T, O_{T})) =
Γ(T, O_{T})^{*}.
`

Let us denote the functor of points of `S` by the
notation `h _{S}`, so that

`
h : SCH _{k} →
FUN(SCH_{k}^{op},
SET).
`

For the functor `h` to be faithful, we must show that
if two morphisms of schemes `f, g : S → T`
induce the same natural transformation `h _{S} →
h_{T}`, then they must have started out as the same morphism.
But this is also easy, since the natural transformation is defined by
composing with the morphism, and we can test it on an identity
morphism.

`
h ^{0} : SCH_{k} →
FUN(ALG_{k},
SET),
`

Comments on this web site should be addressed to the author:

Kevin R. CoombesDepartment of Biomedical Informatics

The Ohio State University

Columbus, Ohio 43210