Algebraic surfaces
2. Affine varieties and projective varieties
0.1. affine varieties. The main object in algebraic geometry is variety. To set it up, let’s first fix an algebraically closed field k. The affine n-space over k, denoted Ank, is the set of all n-tuples. To study An, the polynomial ring A := k[x1, .., xn] is a convenient tool. They are closely connected via the following operation:
(1) Given a set of polynomials T , one can define Z(T ), the common zero locus of T . We call such Z(T ) an algebraic set.
(2) Given a subset Y of affine space, one can define I(Y ) which consists of polynomials vanish along Y . It’s immediate that I(Y ) is an ideal.
These two operation give connection between ideals and algebraic sets. It’s not difficult to see that one can define a topology on An with algebraic sets as closed sets. This topology is called the Zariski topology. Can one also construct a topology on the algebraic side? The answer is yes, with some extra care.
One notices that different ideals might give the same algebraic set, for example, the ideal (x) and (x3) do. Among all ideals defining the same algebraic set, there is a maximal one, the radical ideal. This is a consequence of:
Theorem 0.1 (Hilbert’s Nullstellensatz). Let k be an algebraically closed field and A = k[x1, ..., xn] be the polynomial ring. Let a be an ideal in A, then
I(Z(a)) =√ a.
Corollary 0.2. There is an one-to-one correspondence between algebraic sets and radical ideals. Furthermore, the algebraic set is irreducible (resp. a point) if and only if its ideal is prime (resp. maximal).
Definition 0.3. An algebraic set X is irreducible it if can’t be written as union of two algebraic set in a non-trivial way. More precisely, if X = X1∪ X2 with Xi
being algebraic sets, then either X = X1 or X = X2.
An affine variety is an irreducible algebraic set in An. An open subset of an affine variety is a quasi-affine variety.
Since A is Noetherian, it’s easy to see that every algebraic set can be written as finite union of affine varieties. (This is basically what primary decomposition does). For an algebraic set X, it has the induced Zariski topology. It’s easy to see that it has descending chain condition for closed subsets, i.e. for any sequence Y1⊇ Y2⊇ . . . of closed subsets, there is an integer r such that Yr= Yr+1= .... A topological space is called Noetherian if ithas d.c.c for closed subsets.
With the correspondence in mind, we can define the concept of dimension geo- metrically and algebraically.
Definition 0.4. For a Noetherian topological space X, the dimension of X, denoted dimX, is defined to be the supremum (=maximum) of the length of chain of closed subvarieties.
For an affine variety XinAn, the polynomial functions A restrict to X is nothing but the homomorphism π : A → A/I(X). The ring A/I(X) is called the coordinate ring of X, denoted A(X). One can recover the geometry of X from A(X) by considering Spec(A(X)), which consist of prime ideals in A(X). One can give the Zariski topology on Spec(A(X)) which is closely related to the Zariski topology on X. This is actually the construction of affine scheme. And affine variety can be viewed as a nice affine scheme.
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Exercise 0.5. The coordinate ring of an affine variety is a domain and a finitely generated k-algebra. Conversely, a domain which is a finitely generated k-algebra is a coordinate ring of an affine variety.
One can also similarly define the Krull dimension or simply dimension to be the supremum of length of chain of prime ideals of a ring. It’s easy to see that for an algebraic set X, then
dimX = dimA(X).
However, it’s not trivial to prove that dimAn= n.
0.2. projective varieties. A projective n-space, denoted Pn is defined to be the set of equivalence classes of (n + 1) tuples (a0, ..., an), with not all zero. Where the equivalent relation is (a0, ..., an) ∼ (λa0, ..., λan) for all λ 6= 0. We usually write the equivalence class as [a0, ..., an] or (a0: ... : an).
One can first consider Pn as a quotient of An+1− {(0, ..., 0)}. Let π : An+1− {(0, ..., 0)} → Pn be the quotient map. And we can topologize Pn by the quotient topology of Zariski topology. Then one sees that for a closed set Y ⊂ Pn, π−1(Y ) corresponds to a homogeneous ideal I C k[x0, ..., xn].
We have the similar correspondence between projective algebraic sets and homo- geneous radical ideals. There is an ieal need to be excluded, the irrelevant maximal ideal, (x0, .., xn).
Another important description is to give Pn an open covering of n + 1 copies of An. It follows that every projective variety can be covered by affine varieties.
To this end, we can simply consider
ıj: An → Pn by ıj(a1, ..., an) = [a1, ..., aj−1, 1, aj, .., an].
On the other hand, let Hj be the hyperplane xj = 0 in Pn. Then we have pj: Pn− Hj → An by pj[a0, ..., an] = (a0/aj, ..., aj−1/aj, aj+1/aj, ..., an/aj).
Example 0.6. We have seen that an elliptic curve E can be maps to C2 by the Weierstrass functions. Compose with ı2, we have a map ϕ : E → P2. The defining equation in C2 is y2 = 4x3− g2x − g3. While the defining equation in P2 is the homogenized equation y2z = 4x3− g2xz2− g3z3
In general, the equations in affine spaces and projective spaces are corresponding by homogenization and dehomoenization.
By a variety, we mean affine, quasi-affine, projective or quasi-projective variety.
(More generally, an abstract variety can be defined as an integral separated scheme of finite type over an algebraically closed field k).
