In this article, we will consider algebraic geometry in the projective space. Throughout this article, k denotes an algebraically closed field.
Let . On the set , we consider the equivalence relation:
The projective n-space is the set of equivalence classes under this relation. An element of is denoted by for any representative .
For example .
The above only defines as a set; we need some geometric structure on it for the concept to be meaningful. Throughout this article we will fix , which is graded by degree.
Suppose is homogeneous and . We write if .
Note that the value is in general not well-defined. Indeed if we have two representatives for , then
Despite this, if and only if so the definition is sensible. As in the case of the affine n-space, we will define a correspondence between ideals of B and subsets of .
Let be a graded ideal. Write
For a finite sequence of homogeneous polynomials we also write for where .
Also let .
Prove the following, for any graded ideals and collection of graded ideals of B.
- If is a set of homogeneous generators of , then
In the other direction, we define:
Let be any subset. Then denotes the (graded) ideal of B generated by:
In summary, we defined the following maps.
Zariski Topology of Projective Space
We wish to define the Zariski topology on ; for that let us take subsets of which can be identified with the affine space . Fix ; let
Note that for the same point can be represented by where the i-th coordinate is 1. This gives a bijection . E.g. for n = 2, we have:
Note that for any , the intersection maps to an open subset of via both and . Indeed if i < j then is the set of all satisfying while is the set of all satisfying . Hence, the following is well-defined.
The Zariski topology on is defined by specifying every as an open subset, where obtains the Zariski topology of from .
A projective variety is a closed subspace .
First, we have the following preliminary results.
For any homogeneous , the set is (Zariski) closed in .
It suffices to show that is closed in for each . But , where . The same holds for . ♦
For any , let
be its homogenization.
1. Prove that if , the homogenization of fg is the product of the homogenizations of f and g.
2. Let be the homogenization of a non-constant . Then f is irreducible if and only if F is irreducible. [ Hint: you may find lemma 2 here helpful. ]
The Zariski topology on is consistent with our earlier notions of closed subsets:
A subset is closed under the Zariski topology if and only if for some graded ideal .
(⇐) Suppose for some homogeneous . Since , by lemma 1 this is Zariski closed.
(⇒) Let ; it suffices to show that any is contained in for some homogeneous . Now is contained in some , say without loss of generality. Hence for some . If , then
, where F = homogenization of f. ♦
Let be the projective variety defined by the homogeneous equation . Then
- is cut out from by ;
- is cut out from by ;
- is cut out from by .
Cone of Projective Variety
We wish to prove the bijective correspondence between graded radical ideals of B and closed subsets . For that, we can piggyback on existing results for the affine case.
Let be any subset. The cone of C is
For any non-empty collection of subsets we have
Also, we have:
If is a proper graded ideal then .
In particular, by proposition 1, the cone of a closed is closed in .
Note: if is non-constant homogeneous, then . Now pick a set of homogeneous generators for ; each is non-constant so
This completes the proof. ♦
Furthermore we have:
A non-empty closed subset is of the form for some closed if and only if
When that happens, we call W a closed cone in .
(⇒) is obvious; for (⇐) clearly W = cone(V) for some subset . Let so that . It remains to show is graded, for we would get by lemma 2.
Indeed if , write as a sum of homogeneous components. Then for any and we have which gives
Thus vanish for any , i.e. . ♦
Thus we have the following correspondences:
The top-left column is a bijection by lemma 3; the bottom row is a bijection by Nullstellensatz. In the proof of lemma 3, we also showed that for a closed cone , the ideal is graded. Conversely, if is graded, is the (non-empty) solution set of a collection of graded polynomials; hence it is a closed cone too.
Hence we have a bijection between
- closed subsets , and
- proper homogeneous radical ideals .
The correspondence takes and so
The last piece of the puzzle is the map which takes closed subsets of to homogeneous ideals of B. As an easy exercise, show that
However so we modify our bijection to:
Theorem (Projective Nullstellensatz).
There is a bijection between:
- closed subsets ;
- homogeneous radical ideals such that where is the irrelevant ideal of B.
Prove that if is a homogeneous ideal then is empty if and only if contains a power of .
[ Hint: prove that the radical of 𝔞 is either (1) or B+. ]