Irreducible Subsets of Projective Space
Throughout this article, k is an algebraically closed field.
We wish to consider irreducible closed subsets of . For that we need the following preliminary result.
Lemma 1.
Let
be a graded ring; a proper homogeneous ideal
is prime if and only if:
.
Proof
Suppose is not prime so there exists
such that
. Since
, among all homogeneous components of a pick
of maximum degree such that
; similarly pick
fo b so
.
The degree-(d+e) component of ab is congruent to mod
. Since
is homogeneous we have
, and by the given condition this means
or
, a contradiction. ♦
Exercise A
Prove that a proper homogeneous ideal of a graded ring A is primary if and only if
.
Proposition 1.
Suppose the closed subset
corresponds to the homogeneous radical ideal
,
, where
.
Then V is irreducible if and only if
is prime.
Proof
(⇒) Suppose V is irreducible; let . If
are homogeneous, then
and
are closed subsets of
properly contained in V. Since V is irreducible the following shows
:
.
(⇐) Let where
is prime. Let
be closed subsets with union V. Now write
and
for homogeneous radical ideals
and
. Then
. Since
is a homogeneous radical ideal,
. By exercise B here,
or
. ♦
Corollary 1.
Let
be a non-empty closed subset. Then
is irreducible if and only if
is irreducible.
Proof
By proposition 1, V is irreducible if and only if is prime. But
(from an exercise here) so the result follows. ♦
Quasi-projective Varieties
Recall that a projective variety is a closed subset of some .
Definition.
A quasi-projective variety is an open subset of a projective variety.
This merely defines it as a set; we need a geometric structure on it.
First, let be homogeneous polynomials of the same degree. If
is such that not all
, then we can define a function on an open subset U of
containing
as follows:
.
The map is well-defined: indeed if we can find an open neighbourhood U of
such that
. Also, if we replace projective coordinates
with
, then each
where
so
We write for the resulting function.
Definition.
Let
and
be quasi-projective varieties and
be a function.
We say
is regular at
if there is an open neighbourhood U of
in V such that
for some homogeneous
of the same degree.
We say
is regular if it is regular at every
, in which case we also say
is a morphism of quasi-projective varieties.
From the above definitions, we obtain the category of all quasi-projective varieties and morphisms between them.
Example 1
First consider the case where and
are closed subsets.
E.g., let . A regular map
in the earlier sense can be expressed as a polynomial
, e.g. take
. Via embeddings
and
taking
and
respectively, f can be written in terms of homogeneous coordinates as
since it is the homogenization of the map . This generalizes to an arbitrary regular map of closed subsets
.
Conversely we have:
Lemma 2.
Let
be regular under the new definition. Then there exist polynomials
which represent
.
Proof
We will prove this for the case where V is irreducible.
For each of , let
be projection onto the i-th coordinate. Then
is regular under the new definition, and by proposition 2 here (and its preceding discussion)
can be represented as a polynomial
. Hence we see that
for polynomials
. ♦
Example 2
Take the map given by
Note that the same set of polynomials works globally over the whole of
.
Example 3
Suppose . Let
be the closed subset defined by
. We define a map
as follows
- Outside the point (1 : 1 : 0), take
.
- Outside the point (1 : -1 : 0), take
.
The map agrees outside those two points since due to the equality
.
Isomorphisms
Definition.
Consider the category of all quasi-projective k-varieties, with morphisms defined as above. Two such varieties are said to be isomorphic if they are isomorphic in the category.
A quasi-projective variety is said to be
- projective if it is isomorphic to a closed subset of some
(this generalizes the existing definition of projective varieties);
- affine if it is isomorphic to an affine k-variety (closed subspace of some
);
- quasi-affine if it is isomorphic to an open subset of an affine variety.
Example 4
In example 3 above, we get an isomorphism since we have the reverse map
.
As an exercise, prove that and
.
Definition.
The coordinate ring of a quasi-projective variety V is the set
,
taken to be a k-algebra via point-wise addition and multiplication:
Note
As before, a regular map of quasi-projective varieties induces a ring homomorphism
. By lemma 2, when V is affine
agrees with our earlier version (we proved this in the case where V is irreducible).
Example 5
For each , we have an automorphism
Note that for
. Also
if and only if g is a scalar multiple of the identity matrix, so we get an injective homomorphism
. In fact this is an isomorphism of groups.
E.g. when n = 1, we get the Möbius transformations:
Example 6
We have an isomorphism between the quasi-affine variety and
via the maps
Hence is an affine variety even though it is not closed in
. From the isomorphism we also have:
Example 7
Let . We will show that V is not affine. Indeed consider the injection
which induces
.
The map is injective since V is dense in . Let us show that it is surjective. Suppose
so that
is regular. Write
where
.
By example 6, we have and
. Since
are all dense in V we have injections
and
so that
. It is easy to show that this means
.
Hence induces an isomorphism of the coordinate rings
. If V is affine, by proposition 1 here
would be an isomorphism of varieties, which is a contradiction since
is not surjective.