Пресс-релиз популярных книг

Авторы: 111 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

Книги: 164 А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Э Ю Я

На сайте 111 авторов, 92 книг, 72 статей, 5913 глав.

5. Projective Varieties and Complete Varieties

Back

Throughout this section, k will be an algebraically closed field. Recall that we

defined

Pn = kn+1 \ {origin}/∼,

where (a0, . . . , an) ∼ (b0, . . . , bn) if and only if there exists a c _= 0 in k such

that (a0, . . . , an) = c(b0, . . . , bn). Write (a0 : . . . : an) for the equivalence class

of (a0, . . . , an), and π for the map kn+1 \ {origin}/∼→ Pn. Let Ui be the set of

(a0 : . . . : an) ∈ Pn such that ai _= 0. Then (a0 : . . . : an) _→ ( a0

, . . . , ai−1

, ai+1

, . . . , an

)

is a bijection vi : Ui → kn, and we used these bijections to define the structure of a

ringed space on Pn; specifically, we said that U ⊂ Pn is open if and only if vi(U ∩ Ui)

is open for all i, and that a function f : U → k is regular if and only if (f|U ∩Ui)◦v−1

is regular on vi(U ∩ Ui) for all i.

In this chapter, we shall first derive another description of the topology on Pn,

and then we shall show that the ringed space structure makes Pn into a separated

algebraic variety. A closed subvariety of Pn (or any variety isomorphic to such a

variety) is called a projective variety, and a locally closed subvariety of Pn (or any

variety isomorphic to such a variety) is called a quasi -projective variety. Note that

every affine variety is quasi-projective, but there are many varieties that are not quasiprojective.

We study morphisms between (quasi-) projective varieties. Finally, we

show that a projective variety is “complete”, that is, it has the analogue of a property

that distinguishes compact topological spaces among locally compact spaces.

Projective varieties are important for the same reason compact manifolds are important:

re sults are often simpler when stated for projective varieties, and the “part

at infinity” often plays a role, even when we would like to ignore it. For example,

a famous theorem of Bezout says that a curve of degree m in the projective plane15

intersects a curve of degree n in exactly mn points (counting multiplicities). For affine

curves, one has only an inequality.

Algebraic subsets of Pn. A polynomial F(X0, . . . ,Xn) is said to be homogeneous

of degree d if it is a sum of terms ai0,... ,inXi0

· · ·Xin

n with i0+· · ·+in = d; equivalently,

F(tX0, . . . , tXn) = tdF(X0, . . . ,Xn)

for all t ∈ k. Write k[X0, . . . ,Xn]d for the subspace of k[X0, . . . ,Xn] of polynomials

of degree d. Then

k[X0, . . . ,Xn] =

d≥0

k[X0, . . . ,Xn]d;

that is, each polynomial F can be written uniquely as a sum F =

Fd with Fd of

degree d.

Let P = (a0 : . . . : an) ∈ Pn. Then P can also be written (ca0 : . . . : can) for

any c ∈ k×, and so we can’t speak of the value of a polynomial F(X0, . . . ,Xn) at P.

However, if F is homogeneous, then F(ca0, . . . , can) = cdF(a0, . . . , an), and so it does

make sense to say that F is zero or not zero at P. We define a projective algebraic

set to be the set of common zeros in Pn of a collection of homogeneous polynomials.

15This means that it is defined by a homogeneous polynomial F(X, Y, Z) of degree m.

Algebraic Geometry: 5. Projective Varieties and Complete Varieties 81

Example 5.1. Consider the projective algebraic subset E of P2 defined by the

homogeneous equation

Y 2Z = X3 + aXZ2 + bZ3 (*)

where X3 + aX + b is assumed not to have multiple roots. It consists of the points

(x : y :1) on the affine curve Eaff

Y 2 = X3 + aX + b,

together with the point “at infinity” (0:1:0).

Poincar´e is usually credited (incorrectly!) with showing that E is an algebraic

group, with the group law such that P +Q+R = 0 if and only if P, Q, and R lie on

a straight line. The zero for the group is the point at infinity.

Curves defined by equations of the form (*) are called elliptic curves. They can

also be described as the curves of genus one, or as the abelian varieties of dimension

one.

In the case k = C, for each equation (*), there is a lattice L ⊂ C and a function

℘ (the Weierstrass ℘-function) that is analytic on C − L and doubly periodic for L

(i.e., such that ℘(z + λ) = ℘(z) for all λ ∈ L) such that

℘

_2 = ℘3 + a℘ + b.

The map z _→ (℘(z) : ℘_(z) : 1) : C/L − {0} → P2 is a bijection from C/L − {0}

onto Eaff. This map can be extended to an isomorphism C/L

≈→ E

by sending 0

(0 :1 : 0).

In the case that a, b ∈ Q, we can speak of the zeros of (*) with coordinates in

Q. They also form a group E(Q), which Mordell showed to be finitely generated. It

is easy to compute the torsion subgroup of E(Q), but there is at present no known

algorithm for computing the rank of E(Q). More precisely, there is an “algorithm”

which always works, but which has not been proved to terminate after a finite amount

of time, at least not in general. There is a very beautiful theory surrounding elliptic

curves over Q and other number fields, whose origins can be traced back 1,800 years

to Diophantus. (See my notes on Elliptic Curves for all of this.)

An ideal a ⊂ k[X0, . . . ,Xn] is said to be homogeneous if it contains with any

polynomial F all the homogeneous components of F, i.e., if F ∈ a ⇒ Fd ∈ a, all d.

Such an ideal is generated by homogeneous polynomials (obviously), and conversely,

an ideal generated by a set of homogeneous polynomials is homogeneous. The radical

of a homogeneous ideal is homogeneous, the intersection of two homogeneous ideals

is homogeneous, and a sum of homogeneous ideals is homogeneous.

For a homogeneous ideal a, we write V (a) for the set of common zeros of the

homogeneous polynomials in a — clearly every polynomial in a will then be zero on

V (a). If F1, . . . , Fr are homogeneous generators for a, then V (a) is the set of common

zeros of the Fi. The sets V (a) have similar properties to their namesakes in An :

a ⊂ b ⇒ V (a) ⊃ V (b);

V (0) = Pn; V (a) = ∅ ⇐⇒ rad(a) ⊃ (X0, . . . ,Xn);

V (ab) = V (a ∩ b) = V (a) ∪ V (b);

V (

ai) = ∩V (ai).

82 Algebraic Geometry: 5. Projective Varieties and Complete Varieties

The first statement is obvious. For the second, let V aff(a) be the zero set of a in

kn+1. It is a cone — it contains together with any point P the line through P and

the origin — and

V (a) = (V aff(a) \ (0, . . . , 0))/∼ .

We have V (a) =∅ ⇐⇒ V aff(a) ⊂ {(0, . . . , 0)} ⇐⇒ rad(a) ⊃ (X0, . . . ,Xn), by

the Hilbert Nullstellensatz. The remaining statements can be proved directly, or by

using the relation between V (a) and V aff(a).

Let C be a cone in kn+1; then I(C) is a homogeneous ideal in k[X0, . . . ,Xn],

because

F(ca0, . . . , can) =

cdFd(a0, . . . , an),

and so, if F(ca0, . . . , can) = 0 for all c ∈ k×, we must also have Fd(a0, . . . , an) = 0.

For any S ⊂ Pn, C = π−1(S)∪{origin} is a cone in kn+1, and we define I(S) = I(C).

Proposition 5.2. The maps V and I define a bijection between the set of algebraic

subsets of Pn and the set of homogeneous radical ideals of k[X0, . . . ,Xn], except that V

maps both the ideals (X0, . . . ,Xn) and k[X0, . . . ,Xn] to the empty set. An algebraic

set V in Pn is irreducible if and only if I(V ) is prime; in particular, Pn is irreducible.

Proof. Note that we have bijections

{algebraic subsets of Pn, _= ∅} π−1 →

{closed cones in kn+1, _= {(0, . . . , 0),∅} →I

{homogeneous radical ideals in k[X0, . . . ,Xn], _= (X0, . . . ,Xn), k[X0, . . . ,Xn]} →V

{algebraic subsets of Pn, _= ∅}.

Here the first map sends V to π−1(V )∪{origin}, which is also the closure of π−1(V ),

and the third map is V in the sense of projective geometry. The composite of any

three of these maps is the identity map. Obviously, V is irreducible if and only if the

closure of π−1(V ) is irreducible, which is true if and only if I(V ) is a prime ideal.

The Zariski topology on Pn. The statements above show that projective algebraic

sets are the closed sets for a topology on Pn. In this subsection, we verify that it agrees

with that defined in the first paragraph of this section. For a homogeneous polynomial

F, let

D(F) = {P ∈ Pn | F(P) _= 0}.

Then, just as in the affine case, D(F) is open and the sets of this type form a basis

for the topology of Pn.

With each polynomial f(X1, . . . ,Xn), we associate the homogeneous polynomial

of the same degree

∗

(X0, . . . ,Xn) = Xdeg(f)

0 f

, . . . ,

and with each homogeneous polynomial F(X0, . . . ,Xn) we associate the polynomial

F∗(X1, . . . ,Xn) = F(1,X1, . . . ,Xn).

Algebraic Geometry: 5. Projective Varieties and Complete Varieties 83

Proposition 5.3. For the topology on Pn just defined, each Ui is open, and when

we endow it with the induced topology, the bijection

Ui ↔ An, (a0 : . . . : 1 : . . . : an) ↔ (a0, . . . , ai−1, ai+1, . . . , an)

becomes a homeomorphism.

Proof. It suffices to prove this with i = 0. The set U0 = D(X0), and so it is a

basic open subset in Pn. Clearly, for any homogeneous polynomial F ∈ k[X0, . . . ,Xn],

D(F(X0, . . . ,Xn)) ∩ U0 = D(F(1,X1, . . . ,Xn)) = D(F∗)

and, for any polynomial f ∈ k[X1, . . . ,Xn],

D(f) = D(f

∗

) ∩ U0.

Thus, under U0 ↔ An, the basic open subsets of An correspond to the intersections

with Ui of the basic open subsets of Pn, which proves that the bijection is a homeomorphism.

Remark 5.4. It is possible to use this to give a different proof that Pn is irreducible.

We apply the criterion that a space is irreducible if and only if every

nonempty open subset is dense (see p22). Note that each Ui is irreducible, and that

Ui∩Uj is open and dense in each of Ui and Uj (as a subset of Ui, it is the set of points

(a0 : . . . : 1 : . . . : aj : . . . : an) with aj _= 0). Let U be a nonempty open subset of Pn;

then U ∩ Ui is open in Ui. For some i, U ∩ Ui is nonempty, and so must meet Ui ∩Uj .

Therefore U meets every Uj , and so is dense in every Uj . It follows that its closure is

all of Pn.

We identify An with U0, and examine the closures in Pn of closed subsets of An.

With each ideal a in k[X1, . . . ,Xn], we associate the homogeneous ideal a∗ in

k[X0, . . . ,Xn] generated by {f∗ | f ∈ a}. For a closed subset V of An, set V ∗ = V (a∗)

with a = I(V ).

With each homogeneous ideal a in k[X0,X1, . . . ,Xn], we associate the ideal a∗ in

k[X1, . . . ,Xn] generated by {F∗ | F ∈ a}. When V is a closed subset of Pn, we set

V∗ = V (a∗) with a = I(V ).

Proposition 5.5. (a) For V a closed algebraic subset of An, V ∗ is the closure

of V in Pn, and (V ∗)∗ = V. If V = ∪Vi is the decomposition of V into its irreducible

components, then V ∗ = ∪V ∗

i is the decomposition of V ∗ into its irreducible

components.

(b) For V a closed algebraic subset of Pn, V∗ = V ∩An. If no irreducible component

of V lies in H∞ or contains H∞, then V∗ is a proper subset of An, and (V∗)∗ = V .

Proof. Straightforward.

The hyperplane at infinity. It is often convenient to think of Pn as being An = U0

with a hyperplane added “at infinity”. More precisely, identify the U0 with An. The

complement of U0 in Pn is H∞ = {(0 : a1 : . . . : an) ⊂ Pn}, which can be identified

with Pn−1.

84 Algebraic Geometry: 5. Projective Varieties and Complete Varieties

For example, P1 = A1 ∪H∞ (disjoint union), with H∞ consisting of a single point,

and P2 = A2 ∪ H∞ with H∞ a projective line. Consider the line

aX + bY + 1 = 0

in A2. Its closure in P2 is the line

aX + bY + Z = 0.

It intersects the hyperplane H∞ = V (Z) at the point (−b : a :0 ), which equals

(1 : −a/b :0) when b _= 0. Note that −a/b is the slope of the line aX + bY + 1 = 0,

and so the point at which a line intersects H∞ depends only on the slope of the line:

parallel lines meet in one point at infinity. We can think of the projective plane P2

as being the affine plane A2 with one point added at infinity for each direction in A2.

Similarly, we can think of Pn as being An with one point added at infinity for each

direction in An — being parallel is an equivalence relation on the lines in An, and

there is one point at infinity for each equivalence class of lines.

Note that the point at infinity on the elliptic curve Y 2 = X3 + aX + b is the

intersection of the closure of any vertical line with H∞.

Pn is an algebraic variety. For each i, write Oi for the sheaf on Ui defined by the

bijection An ↔ Ui ⊂ Pn.

Lemma 5.6. Write Uij = Ui ∩ Uj; then Oi|Uij = Oj |Uij . When endowed with this

sheaf Uij is an affine variety; moreover, Γ(Uij ,Oi) is generated as a k-algebra by the

functions (f|Uij)(g|Uij) with f ∈ Γ(Ui,Oi), g ∈ Γ(Uj ,Oj).

Proof. It suffices to prove this for (i, j) = (0, 1). All rings occurring in the proof

will be identified with subrings of the field k(X0,X1, . . . ,Xn).

Recall that

U0 = {(a0 : a1 : . . . : an) | a0 _= 0}; (a0 : a1 : . . . : an) ↔ (

, . . . ,

) ∈ An.

Let k[X1

, X2

, . . . , Xn

] be the subring of k(X0,X1, . . . ,Xn) generated by the quotients

—it is the polynomial ring in the n variables X1

, . . . , Xn

. An element

f(X1

, . . . , Xn

) ∈ k[X1

, . . . , Xn

] defines the map

(a0 : a1 : . . . : an) _→ f(

, . . . ,

) : U0 → k,

and in this way k[X1

, X2

, . . . , Xn

] becomes identified with the ring of regular functions

on U0, and U0 with Specm k[X1

, . . . , Xn

Next consider the open subset of U0,

U01 = {(a0 : . . . : an) | a0 _= 0, a1 _= 0}.

It is D(X1

), and is therefore an affine subvariety of (U0,O0). The inclusion U01 8→

U0 corresponds to the inclusion of rings k[X1

, . . . , Xn

] 8→ k[X1

, . . . , Xn

, X0

]. An

element f(X1

, . . . , Xn

, X0

) of k[X1

, . . . , Xn

, X0

] defines the function (a0 : . . . : an) _→

f( a1

, . . . , an

, a0

) on U01.

Algebraic Geometry: 5. Projective Varieties and Complete Varieties 85

Similarly,

U1 = {(a0 : a1 : . . . : an) | a1 _= 0}; (a0 : a1 : . . . : an) ↔ (

, . . . ,

) ∈ An,

and we identify U1 with Specm k[X0

, X2

, . . . , Xn

]. An element f(X0

, . . . , Xn

) ∈

k[X0

, . . . , Xn

] defines the map (a0 : . . . : an) _→ f( a0

, . . . , an

) : U1 → k.

When regarded as an open subset of U1,

U01 = {(a0 : . . . : an) | a0 _= 0, a1 _= 0},

is D(X0

), and is therefore an affine subvariety of (U1,O1), and the inclusion U01 8→

U1 corresponds to the inclusion of rings k[X0

, . . . , Xn

] 8→ k[X0

, . . . , Xn

, X1

]. An

element f(X0

, . . . , Xn

) of k[X0

, . . . , Xn

, X1

] defines the function (a0 : . . . : an) _→

f( a0

, . . . , an

, a1

) on U01.

The two rings k[X1

, . . . , Xn

, X0

], k[X0

, . . . , Xn

, X1

] are equal as subrings of

k(X0,X1, . . . ,Xn), and an element of this ring defines the same function on U01

regardless of which of the two rings it is considered an element. Therefore,

whether we regard U01 as a subvariety of U0 or of U1 it inherits the same structure

as an affine algebraic variety. This proves the first two assertions, and the

third is obvious: k[X1

, . . . , Xn

, X0

] is generated by its subrings k[X1

, . . . , Xn

] and

k[X0

, X2

, . . . , Xn

Write ui for the map An → Ui ⊂ Pn. For any open subset U of Pn, we define

f : U → k to be regular if and only if f ◦ ui is a regular function on u

−1

i (U) for all i.

This obviously defines a sheaf O of k-algebras on Pn.

Proposition 5.7. For each i, the bijection An → Ui is an isomorphism of ringed

spaces, An → (Ui, O|Ui); therefore (Pn,O) is a prevariety. It is in fact a variety.

Proof. Let U be an open subset of Ui. Then f : U → k is regular if and only if

(a) it is regular on U ∩ Ui, and

(b) it is regular on U ∩ Uj for all j _= i.

But the last lemma shows that (a) implies (b) because U ∩Uj ⊂ Uij . To prove that

Pn is separated, apply the criterion (3.26c) to the covering {Ui} of Pn.

Example 5.8. Assume k does not have characteristic 2, and let C be the plane

projective curve: Y 2Z = X3. For each a ∈ k×, there is an automorphism

ϕa : C → C, (x : y : z) _→ (ax : y : a3z).

Patch two copies of C × A1 together along C × (A1 −{0}) by identifying (P, u) with

(ϕu(P), u−1), P ∈ C, u ∈ A1−{0}. One obtains in this way a singular 2-dimensional

variety that is not quasi-projective (see Hartshorne 1977, p171). (It is even complete

(see below), and so if it were quasi-projective, it would be projective. It is known that

every irreducible separated curve is quasi-projective, and every nonsingular complete

surface is projective, and so this is an example of minimum dimension. In Shafarevich

1994, VI.2.3 there is an example of a nonsingular complete variety of dimension 3 that

is not projective.)

86 Algebraic Geometry: 5. Projective Varieties and Complete Varieties

The field of rational functions of a projective variety. Recall (page 24) that

we attached to each irreducible variety V a field k(V ) with the property that k(V ) is

the field of fractions of k[U] for any open affine U ⊂ V . We now describe this field in

the case that V = Pn. Recall that k[U0] = k[X1

, . . . , Xn

]. We regard this as a subring

of k(X0, . . . ,Xn), and wish to identify the field of fractions of k[U0] as a subfield of

k(X0, . . . ,Xn). Any nonzero F ∈ k[U0] can be written

, . . . ,

) =

F∗(X0, . . . ,Xn)

Xdeg(F)

and it follows that the field of fractions of k[U0] is

k(U0) =

G(X0, . . . ,Xn)

H(X0, . . . ,Xn)

| G, H homogeneous of the same degree

∪ {0}.

Write k(X0, . . . ,Xn)0 for this field (the subscript 0 is short for “subfield of elements

of degree 0”), so that k(Pn) = k(X0, . . . ,Xn)0. Note that an element F = G

H in

k(X0, . . . ,Xn)0 defines a well-defined function

D(H) → k, (a0 : . . . : an) _→ G(a0, . . . , an)

H(a0, . . . , an)

which is obviously regular (look at its restriction to Ui).

We now extend this discussion to any irreducible projective variety V . Such a V

can be written V = V (p), where p is a homogeneous ideal in k[X0, . . . ,Xn]. Let

kh[V ] = k[X0, . . . ,Xn]/p—it is called the homogeneous coordinate ring of V . (Note

that kh[V ] is the ring of regular functions on the affine cone over V ; therefore its

dimension is dim(V ) + 1. It depends, not only on V , but on the embedding of V into

Pn—it is not intrinsic to V (see 5.17 below).) We say that a nonzero f ∈ kh[V ] is

homogeneous of degree d if it can be represented by a homogeneous polynomial F of

degree d in k[X0, . . . ,Xn]. We give 0 degree 0.

Lemma 5.9. Each element of kh[V ] can be written uniquely in the form

f = f0 + · · · + fd

with fi homogeneous of degree i.

Proof. Let F represent f; then F can be written F = F0 + · · · + Fd with Fi

homogeneous of degree i, and when reduced modulo p, this gives a decomposition

of f of the required type. Suppose f also has a decomposition f =

gi, with gi

represented by the homogeneous polynomial Gi of degree i. Then F − G ∈ p, and

the homogeneity of p implies that Fi − Gi = (F − G)i ∈ p. Therefore fi = gi.

It therefore makes sense to speak of homogeneous elements of k[V ]. For such an

element h, we define D(h) = {P ∈ V | h(P) _= 0}.

Since kh[V ] is an integral domain, we can form its field of fractions kh(V ). Define

kh(V )0 = {g

∈ kh(V ) | g and h homogeneous of the same degree} ∪ {0}.

Proposition 5.10. The field of rational functions on V is kh(V )0.

Algebraic Geometry: 5. Projective Varieties and Complete Varieties 87

Proof. Consider V0

df = U0 ∩ V. As in the case of Pn, we can identify k[V0] with

a subring of kh[V ], and then the field of fractions of k[V0] becomes identified with

kh(V )0.

Regular functions on a projective variety. Again, let V be an irreducible projective

variety. Let f ∈ k(V )0, and let P ∈ V. If we can write f = g

h with g and

h homogeneous of the same degree and h(P) _= 0, then we define f(P) = g(P)

h(P). By

g(P) we mean the following:let P = (a0 : . . . : an); represent g by a homogeneous

G ∈ k[X0, . . . ,Xn], and write g(P) = G(a0, . . . , an); this is independent of the choice

of G, and if (a0, . . . , an) is replaced by (ca0, . . . , can), then g(P) is multiplied by

cdeg(g) = cdeg(h). Thus the quotient g(P)

h(P) is well-defined.

Note that we may be able to write f as g

h with g and h homogeneous polynomials

of the same degree in many essentially different ways (because kh[V ] need not be a

unique factorization domain), and we define the value of f at P if there is one such

representation with h(P) _= 0. The value f(P) is independent of the representation

f = g

h (write P = (a0 : . . . : an) = a; if g

h = g_

h_ in kh(V )0, then gh_ = g_h in kh[V ],

which is the ring of regular functions on the affine cone over V ; hence g(a)h_(a) =

g_(a)h(a), which proves the claim).

Proposition 5.11. For each f ∈ k(V )

= kh(V )0, there is an open subset U of

V where f(P) is defined, and P _→ f(P) is a regular function on U. Every regular

function ϕ on an open subset of V is defined by some f ∈ k(V ).

Proof. Straightforward from the above discussion. Note that if the functions

defined by f1 and f2 agree on an open subset of V , then f1 = f2 in k(V ).

Remark 5.12. (a) The elements of k(V) = kh(V )0 should be thought of as the

analogues of meromorphic functions on a complex manifold; the regular functions on

an open subset U of V are the “meromorphic functions without poles” on U. [In fact,

when k = C, this is more than an analogy:a nonsingular projective algebraic variety

over C defines a complex manifold, and the meromorphic functions on the manifold

are precisely the rational functions on the variety. For example, the meromorphic

functions on the Riemann sphere are the rational functions in z.]

(b) We shall see presently (5.19) that, for any nonzero homogeneous h ∈ kh[V ],

D(h) is an open affine subset of V . The ring of regular functions on it is

k[D(h)] = {g/hm | g homogeneous of degree mdeg(h)} ∪ {0}.

We shall also see that the ring of regular functions on V itself is just k, i.e., any

regular function on an irreducible (connected will do) projective variety is constant.

However, if U is an open nonaffine subset of V , then the ring Γ(U,OV) of regular

functions can be almost anything—it needn’t even be a finitely generated k-algebra!

Morphisms from projective varieties. We describe the morphisms from a projective

variety to another variety.

Proposition 5.13. The map

π : An+1 \ {origin} → Pn, (a0, . . . , an) _→ (a0 : . . . : an)

88 Algebraic Geometry: 5. Projective Varieties and Complete Varieties

is an open morphism of algebraic varieties. A map α : Pn → V with V a prevariety

is regular if and only if α ◦ π is regular.

Proof. The restriction of π to D(Xi) is the projection

(a0, . . . , an) _→ (

: . . . :

) : kn+1 \ V (Xi) → Ui,

which is the regular map of affine varieties corresponding to the map of k-algebras

, . . . ,

→ k[X0, . . . ,Xn][X−1

i ].

(In the first algebra Xj

is to be thought of as a single variable.) It now follows from

(3.5) that π is regular.

Let U be an open subset of kn+1 \ {origin}, and let U_ be the union of all the

lines through the origin that meet U, that is, U_ = π−1π(U). Then U_ is again open

in kn+1 \ {origin}, because U_ = ∪cU, c ∈ k×, and x _→ cx is an automorphism of

kn+1 {origin}. The complement Z of U_ in kn+1 {origin} is a closed cone, and

the proof of (5.2) shows that its image is closed in Pn; but π(U) is the complement

of π(Z). Thus π sends open sets to open sets.

The rest of the proof is straightforward.

Thus, the regular maps Pn → V are just the regular maps An+1 {origin} → V

factoring through Pn (as maps of sets).

Remark 5.14. Consider polynomials F0(X0, . . . ,Xm), . . . , Fn(X0, . . . ,Xm) of the

same degree. The map

(a0 : . . . : am) _→ (F0(a0, . . . , am) : . . . : Fn(a0, . . . , am))

obviously defines a regular map to Pn on the open subset of Pm where not all Fi vanish,

that is, on the set ∪D(Fi) = Pn \ V (F1, . . . , Fn). Its restriction to any subvariety V

of Pm will also be regular. It may be possible to extend the map to a larger set by

representing it by different polynomials. Conversely, every such map arises in this

way, at least locally. More precisely, there is the following result.

Proposition 5.15. Let V = V (a) ⊂ Pm, W = V (b) ⊂ Pn. A map

ϕ: V → W is regular if and only if, for every P ∈ V , there exist polynomials

F0(X0, . . . ,Xm), . . . , Fn(X0, . . . ,Xm), homogeneous of the same degree, such that

Q = (b0 : . . . : bn) _→ (F0(b0, . . . , bm) : . . . : Fn(b0, . . . , bm))

for all points Q = (b0 : . . . : bm) in some neighbourhood of P in V (a).

Proof. Straightforward.

Example 5.16. We prove that the circle X2+Y 2 = Z2 is isomorphic to P1. After

an obvious change of variables, the equation of the circle becomes C : XZ = Y 2.

Define

ϕ : P1 → C, (a : b) _→ (a2 : ab : b2).

For the inverse, define

ψ : C → P1 by

(a : b : c) _→ (a : b) ifa _= 0

(a : b : c) _→ (b : c) ifb _= 0

Algebraic Geometry: 5. Projective Varieties and Complete Varieties 89

Note that,

a _= 0 _= b, ac = b2 ⇒ c

and so the two maps agree on the set where they are both defined. Clearly, both ϕ

and ψ are regular, and one checks directly that they are inverse.

Examples of regular maps of projective varieties. We list some of the classic

maps.

Example 5.17. Let L =

ciXi be a nonzero linear form in n+1 variables. Then

the map

(a0 : . . . : an) _→ (

L(a)

, . . . ,

L(a)

)

is a bijection of D(L) ⊂ Pn onto the hyperplane L(X1, . . . ,Xn) = 1 of An+1, with

inverse

(a0, . . . , an) _→ (a0 : . . . : an).

Both maps are regular — for example, the components of the first map are the

regular functions Xj PciXi

. As V (L−1) is affine, so also is D(L), and its ring of regular

functions is k[ X0 PciXi

, . . . , Xn PciXi

]. (This is really a polynomial ring in n variables—any

one variable Xj/

ciXi for which cj _= 0 can be omitted—see lemma 4.11.)

Example 5.18. (The Veronese mapping.) Let

I = {(i0, . . . , in) ∈ Nn+1 |

ij = m}.

Note that I indexes the monomials of degree m in n + 1 variables. It has ( m+n

m )

elements16. Write νn,m = (m+n

m ) − 1, and consider the projective space Pνn,m whose

coordinates are indexed by I; thus a point of Pνn,m can be written (. . . : bi0...in : . . . ).

The Veronese mapping is defined to be

v : Pn → Pνn,m, (a0 : . . . : an) _→ (. . . : bi0...in : . . . ), bi0...in = ai0

0 . . .ain

n .

For example, when n = 1 and m = 2, the Veronese map is

P1 → P2, (a0 : a1) _→ (a20

: a0a1 : a21

Its image is the curve ν(P1) : X0X2 = X2

1, and the map

(b2,0 : b1,1 : b0,2) _→

(b2,0 : b1,1) if b2,0 _= 1

(b1,1 : b0,2) if b0,2 _= 0.

16This can be proved by induction on m + n. If m = 0 = n, then (00

) = 1, which is correct. A

general homogeneous polynomial of degree m can be written uniquely as

F(X0,X1, . . . ,Xn) = F1(X1, . . . ,Xn) + X0F2(X0,X1, . . . ,Xn)

with F1 homogeneous of degree m and F2 homogeneous of degree m− 1. But

(m+n

n ) = (m+n−1

m ) +

_m+n−1

m−1

because they are the coefficients of Xm in

(X + 1)m+n = (X +1)(X +1)m+n−1,

and this proves what we want.

90 Algebraic Geometry: 5. Projective Varieties and Complete Varieties

is an inverse ν(P1) → P1. (Cf. Example 5.17.) 17

When n = 1 and m is general, the Veronese map is

P1 → Pm, (a0 : a1) _→ (am0

: am−1

0 a1 : . . . : am1

I claim that, in the general case, the image of ν is a closed subset of Pνn,m and that

ν defines an isomorphism of projective varieties ν : Pn → ν(Pn).

First note that the map has the following interpretation:i f we regard the coordinates

ai of a point P of Pn as being the coefficients of a linear form L =

aiXi

(well-defined up to multiplication by nonzero scalar), then the coordinates of ν(P)

are the coefficients of the homogeneous polynomial Lm with the binomial coefficients

omitted.

As L _= 0 ⇒ Lm _= 0, the map ν is defined on the whole of Pn, that is,

(a0, . . . , an) _= (0, . . . , 0) ⇒ (. . . , bi0...in, . . . ) _= (0, . . . , 0).

Moreover, L1 _= cL2 ⇒ Lm1

_= cLm2

, because k[X0, . . . ,Xn] is a unique factorization

domain, and so ν is injective. It is clear from its definition that ν is regular.

We shall see later in this section that the image of any projective variety under a

regular map is closed, but in this case we can prove directly that ν(Pn) is defined by

the system of equations:

bi0...inbj0...jn = bk0...knb-0...-n, ih + jh = kh + Bh, all h (*).

Obviously Pn maps into the algebraic set defined by these equations. Conversely, let

Vi = {(. . . . : bi0...in : . . . ) | b0...0m0...0 _= 0}.

Then ν(Ui) ⊂ Vi and ν−1(Vi) = Ui. It is possible to write down a regular map Vi → Ui

inverse to ν|Ui:fo r example, define V0 → Pn to be

(. . . : bi0...in : . . . ) _→ (bm,0,... ,0 : bm−1,1,0,... ,0 : bm−1,0,1,0,... ,0 : . . . : bm−1,0,... ,0,1).

Finally, one checks that ν(Pn) ⊂ ∪Vi.

For any closed variety W ⊂ Pn, ν|W is an isomorphism of W onto a closed subvariety

ν(W) of ν(Pn) ⊂ Pνn,m.

Remark 5.19. The Veronese mapping has a very important property. If F is a

nonzero homogeneous form of degree m ≥ 1, then V (F) ⊂ Pn is called a hypersurface

of degree m and V (F) ∩ W is called a hypersurface section of the projective variety

W. When m = 1, “surface” is replaced by “plane”.

Now let H be the hypersurface in Pn of degree m

ai0...inXi0

· · ·Xin

n = 0,

and let L be the hyperplane in Pνn,m defined by

ai0...inXi0...in.

17Note that, although P1 and ν(P1) are isomorphic, their homogeneous coordinate rings are not.

In fact kh[P1] = k[X0,X1], which is the affine coordinate ring of the smooth variety A2, whereas

kh[ν(P1)] = k[X0,X1,X2]/(X0X2 − X2

1 ) which is the affine coordinate ring of the singular variety

X0X2 − X2

1 .

Algebraic Geometry: 5. Projective Varieties and Complete Varieties 91

Then ν(H) = ν(Pn) ∩ L, i.e.,

H(a) = 0 ⇐⇒ L(ν(a)) = 0.

Thus for any closed subvariety W of Pn, ν defines an isomorphism of the hypersurface

section W ∩H of V onto the hyperplane section ν(W)∩L of ν(W). This observation

often allows one to reduce questions about hypersurface sections to questions about

hyperplane sections.

As one example of this, note that ν maps the complement of a hypersurface section

of W isomorphically onto the complement of a hyperplane section of ν(W), which we

know to be affine. Thus the complement of any hypersurface section of a projective

variety is an affine variety—we have proved the statement in (5.12b).

Example 5.20. An element A = (aij) of GLn+1 defines an automorphism of Pn:

(x0 : . . . : xn) _→ (. . . :

aijxj : . . . );

clearly it is a regular map, and the inverse matrix gives the inverse map. Scalar

matrices act as the identity map.

Let PGLn+1 = GLn+1 /k×I, where I is the identity matrix, that is, PGLn+1 is the

quotient of GLn+1 by its centre. Then PGLn+1 is the complement in P(n+1)2−1 of the

hypersurface det(Xij ) = 0, and so it is an affine variety with ring of regular functions

k[PGLn+1] = {F(. . . ,Xij, . . . )/ det(Xij)m | deg(F) = m · (n + 1)} ∪ {0}.

It is an affine algebraic group.

The homomorphism PGLn+1 → Aut(Pn) is obviously injective. It is also surjective

— see Mumford, Geometric Invariant Theory, Springer, 1965, p20.

Example 5.21. (The Segre mapping.) This is the mapping

((a0 : . . . : am), (b0 : . . . : bn)) _→ ((. . . : aibj : . . . )) : Pm × Pn → Pmn+m+n.

The index set for Pmn+m+n is {(i, j) | 0 ≤ i ≤ m, 0 ≤ j ≤ n}. Note that if we

interprete the tuples on the left as being the coefficients _ of two linear forms L1 =

aiXi and L2 =

bjYj , then the image of the pair is the set of coefficients of the

homogeneous form of degree 2, L1L2. From this observation, it is obvious that the

map is defined on the whole of Pm × Pn (L1 _= 0 _= L2 ⇒ L1L2 _= 0) and is injective.

On any subset of the form Ui × Uj it is defined by polynomials, and so it is regular.

Again one can show that it is an isomorphism onto its image, which is the closed

subset of Pmn+m+n defined by the equations

wijwkl − wilwkj = 0.

(See Shafarevich 1988, I.5.1) For example, the map

((a0 : a1), (b0 : b1)) _→ (a0b0 : a0b1 : a1b0 : a1b1) : P1 × P1 → P3

has image the hypersurface

H : WZ = XY.

The map

(w : x : y : z) _→ ((w : y), (w : x))

92 Algebraic Geometry: 5. Projective Varieties and Complete Varieties

is an inverse on the set where it is defined. [Incidentally, P1 × P1 is not isomorphic

to P2, because in the first variety there are closed curves, e.g., two vertical lines, that

don’t intersect.]

If V and W are closed subvarieties of Pm and Pn, then the Segre map sends V ×W

isomorphically onto a closed subvariety of Pmn+m+n. Thus products of projective

varieties are projective.

There is an explicit description of the topology on Pm ×Pn :the closed sets are the

sets of common solutions of families of equations

F(X0, . . . ,Xm; Y0, . . . , Yn) = 0

with F separately homogeneous in the X’s and in the Y ’s.

Example 5.22. Let L1, . . . , Ln−d be linearly independent linear forms in n + 1

variables; their zero set E in kn+1 has dimension d + 1, and so their zero set in Pn is

a d-dimensional linear space. Define π : Pn − E → Pn−d−1 by π(a) = (L1(a) : . . . :

Ln−d(a)); such a map is called a projection with centre E. If V is a closed subvariety

disjoint from E, then π defines a regular map V → Pn−d−1. More generally, if

F1, . . . , Fr are homogeneous forms of the same degree, and Z = V (F1, . . . , Fr), then

a _→ (F1(a) : . . . : Fr(a)) is a morphism Pn − Z → Pr−1.

By carefully choosing the centre E, it is possible to project any smooth curve in Pn

isomorphically onto a curve in P3, and nonisomorphically (but bijectively on an open

subset) onto a curve in P2 with only nodes as singularities.18 For example, suppose

we have a nonsingular curve C in P3. To project to P2 we need three linear forms

L0, L1, L2 and the centre of the projection is the point where all forms are zero. We

can think of the map as projecting from the centre P0 onto some (projective) plane

by sending the point P to the point where P0P intersects the plane. To project C to

a curve with only ordinary nodes as singularities, one needs to choose P0 so that it

doesn’t lie on any tangent to C, any trisecant (line crossing the curve in 3 points), or

any chord at whose extremities the tangents are coplanar. See for example Samuel,

P., Lectures on Old and New Results on Algebraic Curves, Tata Notes, 1966.

Proposition 5.23. Let V be a projective variety, and let S be a finite set of points

of V. Then S is contained in an open affine subset of V .

Proof. Find a hyperplane passing through at least one point of V but missing

the elements of S, and apply 5.19. (See the exercises.)

Remark 5.24. There is a converse:let V be a nonsingular complete (see below)

irreducible variety; if every finite set of points in V is contained in an open affine

subset of V then V is projective. (Conjecture of Chevalley; proved by Kleiman about

1966.)

Complete varieties. Complete varieties are the analogues in the category of varieties

of compact topological spaces in the category of Hausdorff topological spaces.

Recall that the image of a compact space under a continuous map is compact, and

hence is closed if the image space is Hausdorff. Moreover, a Hausdorff space V is

18A nonsingular curve of degree d in P2 has genus (d−1)(d−2)

2 . Thus, if g is not of this form, a

curve of genus g can’t be realized as a nonsingular curve in P2.

Algebraic Geometry: 5. Projective Varieties and Complete Varieties 93

compact if and only if, for all topological spaces W, the projection q : V × W → W

is closed, i.e., maps closed sets to closed sets (see Bourbaki, Topologie G´en´erale, I,

§10).

Definition 5.25. An algebraic variety V is said to be complete if for all algebraic

varieties W, the projection q : V ×W → W is closed.

Note that a complete variety is required to be separated — we really mean it to be

a variety and not a prevariety.

Example 5.26. Consider the projection

(x, y) _→ y : A1 × A1 → A1

This is not closed; for example, the variety V : XY = 1 is closed in A2 but its image

in A1 omits the origin. However, if we replace V with its closure in P1 ×A1, then its

projection is the whole of A1.

Proposition 5.27. Let V be complete.

(a) A closed subvariety of V is complete.

(b) If V _ is complete, so also is V × V _.

is a subvariety of W, then it is closed in W.

(d) If V is connected, then any regular map α : V → P1 is either constant or onto.

(e) If V is connected, then any regular function on V is constant.

Proof. (a) Let Z be a closed subvariety of a complete variety V . Then for any

variety W, Z × W is closed in V × W, and so the restriction of the closed map

q : V ×W → W to Z ×W is also closed.

(b) The projection V × V _ ×W → W is the composite of the projections

V × V _ ×W → V _ ×W → W,

both of which are closed.

(because W is a variety, see 3.25), and α(V ) is the projection of Γα onto W. Since V

is complete, the projection is closed, and so α(V ) is closed, and hence is a subvariety

of W. Consider

Γα ×W → α(V ) ×W → W.

We have that Γα is complete (because it is isomorphic to V , see 3.25), and so the

mapping Γα × W → W is closed. As Γα → α(V ) is surjective, it follows that

α(V ) ×W → W is also closed.

(d) Recall that the only proper closed subsets of P1 are the finite sets, and such a

set is connected if and only if it consists of a single point. Because α(V ) is connected

and closed, it must either be a single point (and α _____________is constant) or P1 (and α is onto).

(e) A regular function on V is a regular map f : V → A1 ⊂ P1. Regard it as a map

into P1. If it isn’t constant, it must be onto, which contradicts the fact that it maps

into A1.

Corollary 5.28. Consider a regular map α: V → W; if V is complete and

connected and W is affine, then the image of α is a point.

94 Algebraic Geometry: 5. Projective Varieties and Complete Varieties

Proof. Embed W as a closed subvariety of An, and write α = (α1, . . . , αn) where

each αi is a regular map W → A1. Then each αi is a regular function on V , and

hence is constant.

Remark 5.29. The statement that a complete variety V is closed in any larger

varietyW perhaps explains the name:if V is complete,W is irreducible, and dimV =

dimW, then V = W. (Contrast An ⊂ Pn.)

Theorem 5.30. A projective variety is complete.

Lemma 5.31. A variety V is complete if and only if q : V × W → W is a closed

mapping for all irreducible affine varieties W.

Proof. Straightforward.

After (5.27a), it suffices to prove the Theorem for projective space Pn itself; thus

we have to prove that the projection W × Pn → W is a closed mapping in the case

that W is an affine variety. Note that W ×Pn is covered by the open affines W ×Ui,

0 ≤ i ≤ n, and that a subset U of W × Pn is closed if and only if its intersection

with each W × Ui is closed. We shall need another more explicit description of the

topology on W × Pn.

Let A = k[W], and let B = A[X0, . . . ,Xn]. Note that B = A⊗k k[X0, . . . ,Xn], and

so we can view it as the ring of regular functions on W ×An+1: f ⊗g takes the value

f(w) · g(a) at the point (w, a) ∈ W × An+1. The ring B has an obvious grading—

a monomial aXi0

0 . . . Xin

n , a ∈ A, has degree

ij—and so we have the notion of a

homogeneous ideal b ⊂ B. It makes sense to speak of the zero set V (b) ⊂ W ×Pn of

such an ideal. For any ideal a ⊂ A, aB is homogeneous, and V (aB) = V (a) × Pn.

Lemma 5.32. (i) For each homogeneous ideal b ⊂ B, the set V (b) is closed, and

every closed subset of W × Pn is of this form.

(ii) The set V (b) is empty if and only if rad(b) ⊃ (X0, . . . ,Xn).

(iii) If W is irreducible, then W = V (b) for some homogeneous prime ideal b.

Proof. In the case that A = k, we proved all this on pp 90–92, and the same

arguments apply in the present more general situation. For example, to see that

V (b) is closed, apply the criterion stated above.

The set V (b) is empty if and only if the cone V aff(b) ⊂ W × An+1 defined by

b is contained in W × {origin}. But

ai0...inXi0

0 . . .Xin

n , ai0...in

∈ k[W], is zero on

W × {origin} if an only if its constant term is zero, and so

Iaff(W × {origin}) = (X0,X1, . . . ,Xn).

Thus, the Nullstellensatz shows that V (b) = ∅ ⇒ rad(b) = (X0, . . . ,Xn). Conversely,

if XN

∈ b for all i, then obviously V (b) is empty.

For the final statement, note that if V (b) is irreducible, then the closure of its

inverse image in W ×An+1 is also irreducible, and so the ideal of functions zero on it

prime.

Proof of 5.30. Write p for the projection W × Pn → W. We have to show that

Z closed in W × Pn implies p(Z) closed in W. If Z is empty, this is true, and so we

can assume it to be nonempty. Then Z is a finite union of irreducible closed subsets

Algebraic Geometry: 5. Projective Varieties and Complete Varieties 95

Zi of W × Pn, and it suffices to show that each p(Zi) is closed. Thus we may assume

that Z is irreducible, and hence that Z = V (b) with b a prime homogeneous ideal in

B = A[X0, . . . ,Xn].

Note that if p(Z) ⊂ W_, W_ a closed subvariety of W, then Z ⊂ W_ × Pn—we can

then replace W with W_. This allows us to assume that p(Z) is dense in W, and we

now have to show that p(Z) = W.

Because p(Z) is dense in W, the image of the cone V aff(b) under the projection

W ×An+1 → W is also dense in W, and so (see 2.21a) the map A → B/b is injective.

Let w ∈ W:w e shall show that if w /∈ p(Z), i.e., if there does not exist a P ∈ Pn

such that (w, P) ∈ Z, then p(Z) is empty, which is a contradiction.

Let m ⊂ A be the maximal ideal corresponding to w. Then mB+b is a homogeneous

ideal, and V (mB+b) = V (mB)∩V (b) = (w×Pn)∩V (b), and so w will be in the image

of Z unless V (mB+b) _= ∅. But if V (mB+b) = ∅, then mB+b ⊃ (X0, . . . ,Xn)N for

some N (by 5.33b), and so mB + b contains the set BN of homogeneous polynomials

of degree N. Because mB and b are homogeneous ideals,

BN ⊂ mB + b ⇒ BN = mBN + BN ∩ b.

In detail:the first inclusion says that an f ∈ BN can be written f = g+h with g ∈ mB

and h ∈ b. On equating homogeneous components, we find that fN = gN + hN.

Moreover: fN = f; if g =

mibi, mi ∈ m, bi ∈ B, then gN =

mibiN; and hN ∈ b

because b is homogeneous. Together these show f ∈ mBN + BN ∩ b.

Let M = BN/BN ∩ b, regarded as an A-module. The displayed equation says

that M = mM. The argument in the proof of Nakayama’s lemma (4.17) shows that

(1 + m)M = 0 for some m ∈ m. Because A → B/b is injective, the image of 1 + m

in B/b is nonzero. But M = BN/BN ∩ b ⊂ B/b, which is an integral domain, and so

the equation (1 + m)M = 0 implies that M = 0. Hence BN ⊂ b, and so XN

∈ b for

all i, which contradicts the assumption that Z = V (b) is nonempty.

Elimination theory. We have shown that, for any closed subset Z of Pm × W,

the projection q(Z) of Z in W is closed. Elimination theory 19 is concerned with

providing an algorithm for passing from the equations defining Z to the equations

defining q(Z). We illustrate this in one case.

Let P = s0Xm+s1Xm−1+· · ·+sm and Q = t0Xn+t1Xn−1+· · ·+tn be polynomials.

The resultant of P and Q is defined to be the determinant

___________

s0 s1 . . . sm

s0 . . . sm

. . . . . .

t0 t1 . . . tn

t0 . . . tn

. . . . . .

___________

n-rows

m-rows

19Elimination theory became unfashionable several decades ago—one prominent algebraic geometer

went so far as to announce that Theorem 5.30 eliminated elimination theory from mathematics,

provoking Abhyankar, who prefers equations to abstractions, to start the chant “eliminate the eliminators

of elimination theory”. With the rise of computers, it has become fashionable again.

96 Algebraic Geometry: 5. Projective Varieties and Complete Varieties

There are n rows of s’s and m rows of t’s, so that the matrix is (m + n) × (m + n);

all blank spaces are to be filled with zeros. The resultant is a polynomial in the

coefficients of P and Q.

Proposition 5.33. The resultant Res(P,Q) = 0 if and only if

(a) both s0 and t0 are zero; or

(b) the two polyomials have a common root.

Proof. If (a) holds, then certainly Res(P,Q) = 0. Suppose that α is a common

root of P and Q, so that there exist polynomials P1 and Q1 of degrees m − 1 and

n − 1 respectively such that

P(X) = (X − α)P1(X), Q(X) = (X − α)Q1(X).

From these equations we find that

P(X)Q1(X) − Q(X)P1(X) = 0. (*)

On equating the coefficients of Xm+n−1, . . . ,X, 1 in (*) to zero, we find that the

coefficients of P1 and Q1 are the solutions of a system of m + n linear equations in

m + n unknowns. The matrix of coefficients of the system is the transpose of the

matrix





s0 s1 . . . sm

s0 . . . sm

. . . . . .

t0 t1 . . . tn

t0 . . . tn

. . . . . .





The existence of the solution shows that this matrix has determinant zero, which

implies that Res(P,Q) = 0.

Conversely, suppose that Res(P,Q) = 0 but neither s0 nor t0 is zero. Because

the above matrix has determinant zero, we can solve the linear equations to find

polynomials P1 and Q1 satisfying (*). If α is a root of P, then it must also be a root

of P1 or Q. If the former, cancel X − α from the left hand side of (*) and continue.

As deg P1 < deg P, we eventually find a root of P that is not a root of P1, and so

must be a root of Q.

The proposition can be restated in projective terms. We define the resultant of two

homogeneous polynomials

P(X, Y ) = s0Xm + s1Xm−1Y + · · · + smY m, Q(X, Y ) = t0Xn + · · · + tnY n,

exactly as in the nonhomogeneous case.

Proposition 5.34. The resultant Res(P,Q) = 0 if and only if P and Q have a

common zero in P1.

Proof. The zeros of P(X, Y ) in P1 are of the form:

(a) (a :1) with a a root of P(X, 1), or

(b) (1 :0 ) in the case that s0 = 0.

Algebraic Geometry: 5. Projective Varieties and Complete Varieties 97

Thus (5.34) is a restatement of (5.33).

Now regard the coefficients of P and Q as indeterminants. The pairs of polynomials

(P,Q) are parametrized by the space Am+1 × An+1 = Am+n+2. Consider the closed

subset V (P,Q) in Am+n+2×P1. The proposition shows that its projection on Am+n+2

is the set defined by Res(P,Q) = 0. Thus, not only have we shown that the projection

of V (P,Q) is closed, but we have given an algorithm for passing from the polynomials

defining the closed set to those defining its projection.

Elimination theory does this in general. Given a family of polynomials

Pi(T1, . . . , Tm;X0, . . . ,Xn), homogeneous in the Xi, elimination theory gives an algorithm

for finding polynomials Rj(T1, . . . , Tn) such that the Pi(a1, . . . , am;X0, . . . ,Xn)

have a common zero if and only if Rj(a1, . . . , an) = 0 for all j. (Our theorem only

shows that the Rj exist.) See Cox et al. 1992, Chapter 8, Section 5..

Maple can find the resultant of two polynomials in one variable:f or example,

entering “resultant((x + a)5, (x + b)5, x)” gives the answer (−a + b)25. Explanation:

the polynomials have a common root if and only if a = b, and this can happen in 25

ways. Macaulay doesn’t seem to know how to do more.

The rigidity theorem. The paucity of maps between projective varieties has some

interesting consequences. First an observation:f or any point w ∈ W, the projection

map V × W → V defines an isomorphism V × {w} → V with inverse v _→ (v,w) :

V → V ×W (this map is regular because its components are).

Theorem 5.35. Let α : V × W → U be a regular map, and assume that V is

complete, that V and W are irreducible, and that U is separated. If there are points

u0 ∈ U, v0 ∈ V , and w0 ∈ W such that

α(V × {w0}) = {u0} = α({v0} ×W)

then α(V ×W) = {u0}.

Proof. Let U0 be an open affine neighbourhood of u0. Because the projection

map q : V ×W → W is closed, Z df = q(α−1(U −U0)) is closed in W. Note that a point

w of W lies outside Z if and only α(V × {w}) ⊂ U0. In particular w0 ∈ W − Z, and

so W −Z is dense in W. As V ×{w} is complete and U0 is affine, α(V ×{w}) must

be a point whenever w ∈ W − Z:in fact, α(V × {w}) = α(v0, w) = {u0}. Thus α is

constant on the dense subset V × (W − Z) of V ×W, and so is constant.

An abelian variety is a complete connected group variety.

Corollary 5.36. Every regular map α : A → B of abelian varieties is the composite

of a homomorphism with a translation; in particular, a regular map α : A → B

such that α(0) = 0 is a homomorphism.

Proof. After composing α with a translation, we may assume that α(0) = 0.

Consider the map

ϕ : A × A → B, ϕ(a, a

) = α(a + a

) − α(a) − α(a

Then ϕ(A×0) = 0 = ϕ(0×A) and so ϕ = 0. This means that α is a homomorphism.

Corollary 5.37. The group law on an abelian variety is commutative.

98 Algebraic Geometry: 5. Projective Varieties and Complete Varieties

Proof. Commutative groups are distinguished among all groups by the fact that

the map taking an element to its inverse is a homomorphism:i f (gh)−1 = g−1h−1,

then, on taking inverses, we find that gh = hg. Since the negative map, a _→ −a :

A → A, takes the identity element to itself, the preceding corollary shows that it is a

homomorphism.

Projective space without coordinates. Let E be a vector space over k of dimension

n + 1. The set P(E) of lines through zero in E has a natural structure of an

algebraic variety:t he choice of a basis for E defines an bijection P(E) → Pn, and

the inherited structure of an algebraic variety on P(E) is independent of the choice

of the basis. Note that in contrast to Pn, which has n + 1 distinguished hyperplanes,

namely, X0 = 0, . . . ,Xn = 0, no hyperplane in P(E) is distinguished.

One can also define the structure of an algebraic variety on the set Gn+1,r(E) of

r-dimensional subspaces in E. The resulting varieties are called Grassmanians. They

are projective.

Bezout’s theorem. Let V be a hypersurface in Pn (that is, a closed subvariety of

codimension 1). For such a variety, I(V) = (F(X0, . . . ,Xn)) with F a homogenous

polynomial without repeated factors. We define the degree of V to be the degree of

The next theorem is one of the oldest, and most famous, in algebraic geometry.

Theorem 5.38 (Bezout). Let C and D be curves in P2 of degrees m and n respectively.

If C and D have no irreducible component in common, then they intersect

in exactly mn points, counted with appropriate multiplicities.

Proof. Decompose C and D into their irreducible components. Clearly it suffices

to prove the theorem for each irreducible component of C and each irreducible

component of D. We can therefore assume that C and D are themselves irreducible.

We know from (1.22) that C ∩ D is of dimension zero, and so is finite. After a

change of variables, we can assume that a _= 0 for all points (a : b : c) ∈ C ∩ D.

Let F(X, Y,Z) and G(X, Y,Z) be the polynomials defining C and D, and write

F = s0Zm + s1Zm−1 + · · · + sm, G= t0Zn + t1Zn−1 + · · · + tn

with si and tj polynomials in X and Y of degrees i and j respectively. Clearly

sm _= 0 _= tn, for otherwise F and G would have Z as a common factor. Let R

be the resultant of F and G, regarded as polynomials in Z. It is a homogeneous

polynomial of degree mn in X and Y , or else it is identically zero. If the latter

occurs, then for every (a, b) ∈ k2, F(a, b, Z) and G(a, b, Z) have a common zero,

which contradicts the finiteness of C ∩D. Thus R is a nonzero polynomial of degree

mn. Write R(X, Y ) = XmnR∗( Y

X) where R∗(T ) is a polynomial of degree ≤ mn in

T = Y

X .

Suppose first that degR∗ = mn, and let α1, . . . , αmn be the roots of R∗ (some of

them may be multiple). Each such root can be written αi = bi

, and R(ai, bi) = 0.

According to (5.34) this means that the polynomials F(ai, bi, Z) and G(ai, bi, Z) have

a common root ci. Thus (ai : bi : ci) is a point on C ∩ D, and conversely, if (a : b : c)

is a point on C ∩ D (so a _= 0), then b

a is a root of R∗(T ). Thus we see in this case,

Algebraic Geometry: 5. Projective Varieties and Complete Varieties 99

that C ∩D has precisely mn points, provided we take the multiplicity of (a : b : c) to

be the multiplicity of b

a as a root of R∗.

Now suppose that R∗ has degree r < mn. Then R(X, Y ) = Xmn−rP(X, Y ) where

P(X, Y ) is a homogeneous polynomial of degree r not divisible by X. Obviously

R(0,1) = 0, and so there is a point (0 : 1 : c) in C ∩ D, in contradiction with our

assumption.

Remark 5.39. The above proof has the defect that the notion of multiplicity has

been too obviously chosen to make the theorem come out right. It is possible to show

that the theorem holds with the following more natural definition of multiplicity. Let

P be an isolated point of C ∩ D. There will be an affine neighbourhood U of P and

regular functions f and g on U such that C ∩ U = V (f) and D ∩ U = V (g). We

can regard f and g as elements of the local ring OP , and clearly rad(f, g) = m, the

maximal ideal in OP . It follows that OP /(f, g) is finite-dimensional over k, and we

define the multiplicity of P in C ∩ D to be dimk(OP /(f, g)). For example, if C and

D cross transversely at P, then f and g will form a system of local parameters at P

— (f, g) = m — and so the multiplicity is one.

The attempt to find good notions of multiplicities in very general situations has

motivated much of the most interesting work in commutative algebra over the last 20

years.

Hilbert polynomials (sketch). Recall that for a projective variety V ⊂ Pn,

kh[V ] = k[X0, . . . ,Xn]/b = k[x0, . . . , xn],

where b = I(V ). We observed that b is homogeneous, and therefore kh[V ] is a graded

ring:

kh[V ] = ⊕m≥0kh[V ]m,

where kh[V ]m is the subspace generated by the monomials in the xi of degree m.

Clearly kh[V ]m is a finite-dimensional k-vector space.

Theorem 5.40. There is a unique polynomial P(V, T) such that P(V,m) =

dimk k[V ]m for all m sufficiently large.

Proof. Omitted.

Example 5.41. For V = Pn, kh[V] = k[X0, . . . ,Xn], and (see the footnote on

page 89), dimkh[V ]m = (m+n

n ) = (m+n)···(m+1)

n! , and so

P(Pn, T) = (T+n

n ) =

(T + n) · · · (T + 1)

The polynomial P(V, T) in the theorem is called the Hilbert polynomial of V .

Despite the notation, it depends not just on V but also on its embedding in projective

space.

Theorem 5.42. Let V be a projective variety of dimension d and degree δ; then

P(V, T) =

T d + terms of lower degree.

Proof. Omitted.

100 Algebraic Geometry: 5. Projective Varieties and Complete Varieties

The degree of a projective variety is the number of points in the intersection of the

variety and of a general linear variety of complementary dimension (see later).

Example 5.43. Let V be the image of the Veronese map

(a0 : a1) _→ (ad0

: ad−1

0 a1 : . . . : ad1

) : P1 → Pd.

Then kh[V ]m can be identified with the set of homogeneous polynomials of degree

m · d in two variables (look at the map A2 → Ad+1 given by the same equations),

which is a space of dimension dm + 1, and so

P(V, T) = dT + 1.

Thus V has dimension 1 (which we certainly knew) and degree d.

Macaulay knows how to compute Hilbert polynomials.

References: Hartshorne 1977, I.7; Atiyah and Macdonald 1969, Chapter 11; Harris

1992, Lecture 13.

Algebraic Geometry: 6. Finite Maps 101

Пресс-релиз популярных книг

5. Projective Varieties and Complete Varieties

Популярные книги

Популярные статьи

Навигация