II.4 Projective Varieties
There are several reasons why it is useful to consider the so calledprojective geometry.
One of them is, for instance, to be able to grasp the “well-known” phenomenon that two affine lines (algebraic sets given by linear equations) intersect at “infinity” (cf.
[Kun85, Ch. I, proposition 5.2, p. 30]). As in the affine case, we want to define the n-dimensional projective space (over k)
Pn:=Pn(K) :={P = (y0 :· · ·:yn)|yi ∈K, at least one yi is nonzero}, where (y0 :· · ·:yn) denotes the equivalence class of the following equivalence relation:
(y0:· · ·:yn)∼(z0 :· · ·:zn) :⇐⇒ ∃λ∈K such that∀i:yi =λzi. By itsk-rational points we mean the set
Pn(k) :={P = (y0 :· · ·:yn)∈Pn| ∃λ∈K such that∀i:λyi ∈k}.
Recall that a polynomialF ∈ k[Y0, . . . , Yn] is called homogeneous of degree d, if it is the sum of monomials of the same degreed, which is equivalent to saying that
F(λY0, . . . , λYn) =λdF(Y0, . . . , Yn) for all λ∈K.
In particular, it makes sense to speak of azeroP = (y0:· · ·:yn)∈Pnof a homogeneous polynomialF of degree d:
F(P) = 0 :⇐⇒ F(y0, . . . , yn) = 0.
This expression is well-defined. An idealI ⊆k[Y0, . . . , Yn] is homogeneous, if it is gen-erated by homogeneous polynomials. Now, for a homogeneous ideal I ⊆k[Y0, . . . , Yn], we call
VI :={P ∈Pn|F(P) = 0 for all F ∈I}
a projective algebraic set (over k). For such an algebraic set V, we define I(V /k) as the homogeneous ideal ofk[Y0, . . . , Yn] generated by
{F ∈k[Y0, . . . , Yn]|F is homogeneous and F(P) = 0 for allP ∈V},
called the (homogeneous) ideal of V (over k). Sometimes, we write I(V) instead of I(V /k), if the context is clear.
Similarly to the affine case, we have the projective Nullstellensatz:
Theorem II.4.1. The map V 7→ I(V) induces an inclusion-reversing one-to-one cor-respondence between the set of all projective algebraic setsV and the set of all homo-geneous radical idealsI ⊆k[Y0, . . . , Yn] that are contained in (Y0, . . . , Yn). The inverse mapping is given by the formation of the zero set. Furthermore, for any homogeneous
idealI 6=k[Y0, . . . , Yn] we have
I(VI) =√ I.
Projective varieties correspond to homogeneous prime ideals and projective points cor-respond to homogeneous maximal ideals.
Proof. This is [Kun85, Ch. I, proposition 5.9, p. 34].
One gets a natural topology on Pn, again called the Zariski topology (over k), by considering the projective algebraic sets as closed. A projective algebraic set is called a projective variety, if it is irreducible in the Zariski topology. Proposition II.1.5 and lemma II.1.6 can easily be proven in the projective case. The dimension and the coordinate ring of a projective algebraic set are defined as in the affine case. But these are not the only relations there are between the projective and the affine geometry. We want to establish one with respect to polynomials:
Letf ∈k[X1, . . . , Xn] be of degree d. We define its homogenization with respect to Yi (for ani∈ {0, . . . , n}) by
fi∗ :=Yidf(Y0
Yi, . . . ,Yi−1
Yi ,Yi+1
Yi , . . . ,Yn
Yi).
This is clearly a homogeneous polynomial of degreedink[Y0, . . . , Yn].
Conversely, letF ∈k[Y0, . . . , Yn] be homogeneous of degreed. We define its deho-mogenization with respect toYi by
F∗i:=F(X1, . . . , Xi,1, Xi+1, . . . , Xn).
This is clearly a polynomial ink[X1, . . . , Xn].
It is immediately seen that those two processes are inverses of each other. Put Hi := V(Yi) and consider the open sets Ui := Pn \Hi for every i = 0, . . . , n. The homogenization process induces the map
ϕi :Ui →An,(y0:. . .:yn)7→(y0
yi, . . . ,yi−1
yi ,yi+1
yi , . . . ,yn
yi). (II.6) Proposition II.4.2. ϕi is a homeomorphism ofUi (with its induced topology) to An (with its Zariski topology). Its inverse map is denoted byφi and is explicitly given by
φi :An→Ui,(x1, . . . , xn)7→(x1 :. . .:xi : 1 :xi+1 :. . .:xn).
Proof. The proof can be found in [Har77, Ch. I, proposition 2.2, p. 10]. There, the ground fieldkis algebraically closed, but this restriction is not needed in the proof.
Since the setsU0, . . . , Unare sort of natural and cover the whole projective spacePn, we call the coveringstandard. Furthermore, we denoteϕi(V∩Ui) byVifor a projective closed setV (we often simply write V ∩An forVi and a fixed iin mind). It is a closed affine set and its ideal consists of all dehomogenized elements (with respect to Yi) of the ideal of V. Conversely, let VI be an affine closed set. We can then speak of the
II.4. PROJECTIVE VARIETIES 27 projective closureVI of VI as the closed projective set defined by the idealI genereated by{fi∗ |f ∈I} for a fixed embeddingφi of VI intoPn.
Proposition II.4.3. 1. IfV is an affine variety, thenV is a projective variety with V =V ∩An.
2. IfV is a projective variety, then V ∩An is an affine variety and eitherV ∩An=∅orV =V ∩An.
There is at least oneisuch thatV ∩Ui is nonempty. We call it anonempty affine part of V and denote it by Va (with a fixediin mind).
Proof. This is [Sil86, Ch. I, proposition 2.6, p. 13].
LetV be a projective variety and let Va∈An be a nonempty affine part ofV. We define
k(V) :=k(Va) (II.7)
asthe function field of V (over k). By proposition II.4.3(2) we have thatVa=V and the following proposition tells us thatk(V) is well-defined.
Proposition II.4.4. LetV be the projective closure of a nonempty affine variety V. Then there is a k-algebra isomorphismk(V) ∼=k(V). In particular, k(V) is a finitely generated extension field ofkof transcendence degree dim(V) = dim(V).
Proof. See [Kun85, Ch. III, proposition 2.13, p. 70].
The last part of this proposition turns out to be quite useful, in the case where V is given by a principal prime ideal, as we have the following criterion for V to be absolutely irreducible (cf. proposition II.1.12):
Proposition II.4.5. 1. IfV =Vf is an affine algebraic set, given byf ∈k[X1, . . . , Xn], then:
V is absolutely irreducible ⇐⇒ k is algebraically closed ink(V).
2. If V = VF is a projective algebraic set, given by a homogeneous polynomial F ∈k[Y0, . . . , Yn], then:
V is absolutely irreducible ⇐⇒ k is algebraically closed ink(V).
Proof. 1. By remark II.1.8 and the fact that I(V /K) is just f ·K[X1, . . . , Xn] we have:
V is absolutely irreducible ⇐⇒ f isabsolutely irreducible,
i.e. irreducible over K. The proposition then is simply [Sti93, Ch. III, Corollary 6.7, p. 108].
2. Similarly to (1). Note that remark II.1.8 obviously holds in the projective case as well.
Our next aim is to generalize the notions of morphisms, rational maps and regular maps of affine algebraic sets to projective algebraic sets.
Definition II.4.6. Letϕi be the map defined in (II.6) with inverse ϕ−1i =φi for every i= 0, . . . , n. We call the map rij := ϕj◦φi :An → An the (i, j)-transition map. Its inverse isrji.
Definition II.4.7. LetV ⊆PnandW ⊆Pm be projective algebraic sets. Letψ:V → W be a mapping such that
1. V =Sn
i=1Vi where Vi are the affine parts ofV.
2. ψi := ψ|Vi is an affine morphism of Vi to an affine part Wi of W (say ψi = (f1i, . . . , fmi ) for fki ∈k[Vi] for every affine partVi).
3. rij(f1i(P), . . . , fmi (P)) = (f1j(P), . . . , fmj(P))∈Wj.
Then ψ is called a (projective) morphism from V to W (over k). The set of all such morphisms is denoted by Mork(V, W).
Example II.4.8. By the above definition of projective morphisms it is immediate that the homeomorphismϕi defined in (II.6) is in fact an isomorphism in the sense of definition II.4.7. Consider the following result on the dimension of a projective variety V:
Proposition II.4.9. LetV ⊆Pnbe a projective variety and letVa⊆Anbe a nonempty affine part ofV. Then:
dimV = dimVa. Proof. This is [Kun85, Ch. II, proposition 4.1, p. 59].
Clearly,An is a nonempty affine part ofPn and so
dimPn= dimAn= dim(k[X1, . . . , Xn]) = trk(k(X1, . . . , Xn)) =n by proposition II.2.3 together with (II.1). SoPn has dimensionn.
A (projective) rational map from a projective variety V ⊆ Pn to A1 is defined as the equivalence class of a rational map defined on the affine parts ofV compatible with the transition maps on intersections of standard affine pieces Ui. The generalization to rational maps that map to projective varieties is then done in the same way as in section II.3. A rational map fromV toP1 is called arational function of V.
LetP ∈Pn be a point,V a projective variety andVi an affine part of it containing P. A rational mapr fromV toW ⊆Pm is called regular at P, if there exists an open neighborhoodU ofP inVi such thatr|Vi is defined onU.
II.5. NONSINGULAR VARIETIES 29