10. Divisors and Intersection Theory

Back

In this section, k is an arbitrary field.

Divisors. Recall that a normal ring is an integral domain that is integrally closed

in its field of fractions. A variety V is normal if Ov is a normal ring for all v ∈ V .

Equivalent condition:fo r every open connected affine subset U of V , Γ(U,OV ) is a

normal ring.

Remark 10.1. Let V be a projective variety, say defined by a homogeneous ring

R. If R is normal, then V is said to be projectively normal. A projectively normal

variety is normal, but the converse statement is false.

Assume now that V is normal and irreducible.

A prime divisor on V is an irreducible subvariety of V of codimension 1. A divisor

on V is an element of the free abelian group Div(V ) generated by the prime divisors.

Thus a divisor D can be written uniquely as a finite (formal) sum

D =

_

niZi, ni ∈ Z, Zi a prime divisor on V.

The support |D| of D is the union of the Zi corresponding to nonzero ni’s. A divisor

is said to be effective (or positive) if ni ≥ 0 for all i. We get a partial ordering on the

divisors defining D ≥ D_ to mean D − D_ ≥ 0.

Because V is normal, there is associated with every prime divisor Z on V a discrete

valuation ring OZ. This can be defined, for example, by choosing an open affine

subvariety U of V such that U ∩Z _= ∅; then U ∩Z is a maximal proper closed subset

of U, and so the ideal p corresponding to it is minimal among the nonzero ideals of

R = Γ(U,O); so Rp is a normal ring with exactly one nonzero prime ideal pR — it is

therefore a discrete valuation ring (Atiyah and MacDonald 9.2), which is defined to

be OZ. More intrinsically we can define OZ to be the set of rational functions on V

that are defined an open subset U of V with U ∩ Z _= ∅.

Let ordZ be the valuation of k(V )× _ Z with valuation ring OZ. The divisor of a

nonzero element f of k(V ) is defined to be

div(f) =

_

ordZ(f) · Z.

The sum is over all the prime divisors of V , but in fact ordZ(f) = 0 for all but finitely

many Z’s. In proving this, we can assume that V is affine (because it is a finite union

of affines), say V = Specm(R). Then k(V ) is the field of fractions of R, and so we

can write f = g/h with g, h ∈ R, and div(f) = div(g) − div(h). Therefore, we can

assume f ∈ R. The zero set of f, V (f) either is empty or is a finite union of prime

divisors, V = ∪Zi (see 7.2) and ordZ(f) = 0 unless Z is one of the Zi.

The map

f _→ div(f) : k(V )

× → Div(V )

is a homomorphism. A divisor of the form div(f) is said to be principal, and two

divisors are said to be linearly equivalent, denoted D ∼ D_ , if they differ by a principal

divisor.

When V is nonsingular, the Picard group Pic(V ) of V is defined to be the group

of divisors on V modulo principal divisors. (Later, we shall define Pic(V ) for an

138 Algebraic Geometry: 10. Divisors and Intersection Theory

arbitrary variety; when V is singular it will differ from the group of divisors modulo

principal divisors, even when V is normal.)

Example 10.2. Let C be a nonsingular affine curve corresponding to the affine

k-algebra R. Because C is nonsingular, R is a Dedekind domain. A prime divisor on

C can be identified with a nonzero prime divisor in R, a divisor on C with a fractional

ideal, and Pic(C) with the ideal class group of R.

Let U be an open subset of V , and let Z be a prime divisor of V. Then Z ∩ U

is either empty or is a prime divisor of U. We define the restriction of a divisor

D =

_

nZZ on V to U to be

D|U =

_

Z∩U_=∅

nZ · Z ∩ U.

When V is nonsingular, every divisor D is locally principal, i.e., every point P has

an open neighbourhood U such that the restriction of D to U is principal. It suffices

to prove this for a divisor Z. If P is not in the support of D, we can take f = 1. The

prime divisors passing through P are in one-to-one correspondence with the prime

ideals p of height 1 in OP , i.e., the minimal nonzero prime ideals. Our assumption

implies that OP is a regular local ring. It is a (fairly hard) theorem in commutative

algebra that a regular local ring is a unique factorization domain. It is a (fairly easy)

theorem that a Noetherian integral domain is a unique factorization domain if every

prime ideal of height 1 is principal (Nagata 1962, 13.1). Thus p is principal in O

p,

and this implies that it is principal in Γ(U,OV ) for some open affine set U containing

P (see also 7.13).

If D|U = div(f), then we call f a local equation for D on U.

Intersection theory. Fix a nonsingular variety V of dimension n over a field k,

assumed to be perfect. Let W1 and W2 be irreducible closed subsets of V , and let

Z be an irreducible component of W1 ∩ W2. Then intersection theory attaches a

multiplicity to Z. We shall only do this in an easy case.

Divisors. Let V be a nonsingular variety of dimension n, and let D1, . . . ,Dn be

effective divisors on V. We say that D1, . . . ,Dn intersect properly at P ∈ |D1|∩. . .∩

|Dn| if P is an isolated point of the intersection. In this case, we define

(D1 · . . . · Dn)P = dimk OP /(f1, . . . , fn)

where fi is a local equation for Di near P. The hypothesis on P implies that this is

finite.

Example 10.3. In all the examples, the ambient variety is a surface.

(a) Let Z1 be the affine plane curve Y 2 − X3, let Z2 be the curve Y = X2, and let

P = (0, 0). Then

(Z1 · Z2)P = dimk[X, Y ](X,Y )/(Y − X3, Y 2 − X3) = dimk[X]/(X4 − X3) = 3.

(b) If Z1 and Z2 are prime divisors, then (Z1 · Z2)P = 1 if and only if f1, f2 are local

uniformizing parameters at P. Equivalently, (Z1 · Z2)P = 1 if and only if Z1 and Z2

are transversal at P, that is, TZ1(P) ∩ TZ2(P) = {0}.

Algebraic Geometry: 10. Divisors and Intersection Theory 139

(c) Let D1 be the x-axis, and let D2 be the cuspidal cubic Y 2−X3. For P = (0, 0),

(D1 · D2)P = 3.

(d) In general, (Z1 · Z2)P is the “order of contact” of the curves Z1 and Z2.

We say that D1, . . . ,Dn intersect properly if they do so at every point of intersection

of their supports; equivalently, if |D1| ∩ . . . ∩ |Dn| is a finite set. We then define the

intersection number

(D1 · . . . · Dn) =

_

P∈|D1|∩...∩|Dn|

(D1 · . . . · Dn)P .

Example 10.4. Let C be a curve. If D =

_

niPi, then the intersection number

(D) =

_

ni[k(Pi) : k].

By definition, this is the degree of D.

Consider a regular map α: W → V of connected nonsingular varieties, and let D

be a divisor on V whose support does not contain the image of W. There is then a

unique divisor α∗D on W with the following property:if D has local equation f on

the open subset U of V , then α∗D has local equation f ◦ α on α−1U. (Use 7.2 to

see that this does define a divisor on W; if the image of α is disjoint from |D|, then

α∗D = 0.)

Example 10.5. Let C be a curve on a surface V, and let α: C_ → C be the

normalization of C. For any divisor D on V ,

(C · D) = degα

∗

D.

Lemma 10.6 (Additivity). Let D1, . . . ,Dn,D be divisors on V . If (D1 · . . . ·Dn)P

and (D1 · . . . · D)P are both defined, then so also is (D1 · . . . · Dn + D)P , and

(D1 · . . . · Dn + D)P = (D1 · . . . · Dn)P + (D1 · . . . · D)P .

Proof. One writes some exact sequences. See Shafarevich 1994, IV.1.2.

Note that in intersection theory, unlike every other part of mathematics, we add

first, and then multiply.

Since every divisor is the difference of two effective divisors, Lemma 10.1 allows us

to extend the definition of (D1 · . . . ·Dn) to all divisors intersecting properly (not just

effective divisors).

Lemma 10.7 (Invariance under linear equivalence). Assume V is complete. If

Dn ∼ D_

n, then

(D1 · . . . · Dn) = (D1 · . . . · D

_

n).

Proof. By additivity, it suffices to show that (D1 · . . .·Dn) = 0 if Dn is a principal

divisor. For n = 1, this is just the statement that a function has as many poles as

zeros (counted with multiplicities). Suppose n = 2. By additivity, we may assume

that D1 is a curve, and then the assertion follows from Example 10.5 because

D principal ⇒ α

∗

D principal.

The general case may be reduced to this last case (with some difficulty). See

Shafarevich 1994, IV.1.3.

140 Algebraic Geometry: 10. Divisors and Intersection Theory

Lemma 10.8. For any n divisors D1, . . . ,Dn on an n-dimensional variety, there

exists n divisors D_

1, . . . ,D_

n intersect properly.

Proof. See Shafarevich 1994, IV.1.4.

We can use the last two lemmas to define (D1 · . . . · Dn) for any divisors on a

complete nonsingular variety V :ch oose D_

1, . . . ,D_

n as in the lemma, and set

(D1 · . . . · Dn) = (D_

1

· . . . · D_

n).

Example 10.9. Let C be a smooth complete curve over C, and let α: C → C be

a regular map. Then the Lefschetz trace formula states that

(Δ · Γα) = Tr(α|H0(C,Q)−Tr(α|H1(C,Q)+Tr(α|H2(C,Q).

In particular, we see that (Δ · Δ) = 2 − 2g, which may be negative, even though Δ

is an effective divisor.

Let α: W → V be a finite map of irreducible varieties. Then k(W) is a finite

extension of k(V ), and the degree of this extension is called the degree of α. If k(W)

is separable over k(V ) and k is algebraically closed, then there is an open subset U

of V such that α−1(u) consists exactly d = deg α points for all u ∈ U. In fact, α−1(u)

always consists of exactly deg α points if one counts multiplicities. Number theorists

will recognize this as the formula

_

eifi = d. Here the fi are 1 (if we take k to be

algebraically closed), and ei is the multiplicity of the ith point lying over the given

point.

A finite map α: W → V is flat if every point P of V has an open neighbourhood

U such that Γ(α−1U,OW) is a free Γ(U,OV )-module — it is then free of rank deg α.

Theorem 10.10. Let α: W → V be a finite map between nonsingular varieties.

For any divisors D1, . . . ,Dn on V intersecting properly at a point P of V ,

_

α(Q)=P

(α∗D1 · . . . · α∗Dn) = degα · (D1 · . . . · Dn)P .

Proof. After replacing V by a sufficiently small open affine neighbourhood of P,

we may assume that α corresponds to a map of rings A → B and that B is free of rank

d = degα as an A-module. Moreover, we may assume that D1, . . . ,Dn are principal

with equations f1, . . . , fn on V , and that P is the only point in |D1| ∩ . . . ∩ |Dn|.

Then mP is the only ideal of A containing a = (f1, . . . , fn). Set S = A \ mP; then

S

−1A/S

−1a = S

−1(A/a) = A/a

because A/a is already local. Hence

(D1 · . . . · Dn)P = dimA/(f1, . . . , fn).

Similarly,

(α∗D1 · . . . · α∗Dn)P = dimB/(f1 ◦ α, . . . , fn ◦ α).

But B is a free A-module of rank d, and

A/(f1, . . . , fn) ⊗A B = B/(f1 ◦ α, . . . , fn ◦ α).

Algebraic Geometry: 10. Divisors and Intersection Theory 141

Therefore, as A-modules, and hence as k-vector spaces,

B/(f1 ◦ α, . . . , fn ◦ α) ≈ (A/(f1, . . . , fn))d

which proves the formula.

Example 10.11. Assume k is algebraically closed of characteristic p _= 0. Let

α: A1 → A1 be the Frobenius map c _→ cp. It corresponds to the map k[X] → k[X],

X _→ Xp, on rings. Let D be the divisor c. It has equation X − c on A1, and α∗D

has the equation Xp − c = (X − γ)p. Thus α∗D = p(γ), and so

deg(α

∗

D) = p = p · deg(D).

The general case. Let V be a nonsingular connected variety. A cycle of codimension

r on V is an element of the free abelian group Cr(V ) generated by the prime cycles

of codimension r.

Let Z1 and Z2 be prime cycles on any nonsingular variety V , and let W be an

irreducible component of Z1 ∩ Z2. We know that

dim Z1 + dim Z2 ≤ dim V + dim W,

and we say Z1 and Z2 intersect properly at W if equality holds.

Define OV,W to be the set of rational functions on V that are defined on some open

subset U of V with U ∩W _= ∅ — it is a local ring. Assume that Z1 and Z2 intersect

properly at W, and let p1 and p2 be the ideals in OV,W corresponding to Z1 and Z2

(so pi = (f1, f2, ..., fr) if the fj define Zi in some open subset of V meeting W). The

example of divisors on a surface suggests that we should set

(Z1 · Z2)W = dimk OV,W/(p1, p2),

but examples show this is not a good definition. Note that

OV,W/(p1, p2) = OV,W/p1 ⊗OV,W

OV,W/p2.

It turns out that we also need to consider the higher Tor terms. Set

χ

O

(O/p1,O/p2) =

d_imV

i=0

(−1)i dimk(Tor

O

i (O/p1,O/p2)

where O = OV,W. It is an integer ≥ 0, and = 0 if Z1 and Z2 do not intersect

properly at W. When they do intersect properly, we define (Z1 · Z2)W = mW,

m = χO(O/p1,O/p2). When Z1 and Z2 are divisors on a surface, the higher Tor’s

vanish, and so this definition agrees with the previous one.

Now assume that V is projective. It is possible to define a notion of rational equivalence

for cycles of codimension r:let W be an irreducible subvariety of codimension

r−1, and let f ∈ k(W)×; then div(f) is a cycle of codimension r on V (since W may

not be normal, the definition of div(f) requires care), and we let Cr(V )_ be the subgroup

of Cr(V ) generated by such cycles as W ranges over all irreducible subvarieties

of codimension r − 1 and f ranges over all elements of k(W)×. Two cycles are said

to be rationally equivalent if they differ by an element of Cr(V )_, and the quotient of

Cr(V ) by Cr(V )_ is called the Chow group CHr(V ). A discussion similar to that in

the case of a surface leads to well-defined pairings

CHr(V ) × CHs(V ) → CHr+s(V ).

142 Algebraic Geometry: 10. Divisors and Intersection Theory

In general, we know very little about the Chow groups of varieties — for example,

there has been little success at algebraic cycles on varieties other than the obvious

one (divisors, intersections of divisors,...).

We can restate our definition of the degree of a variety in Pn as follows:a closed

subvariety V of Pn of dimension d has degree (V ·H) for any linear subspace of Pn of

codimension d. (All linear subspaces of Pnof codimension r are rationally equivalent,

and so (V · H) is independent of the choice of H.)

Remark 10.12. (Bezout’s theorem). A divisor D on Pn is linearly equivalent of

δH, where δ is the degree of D and H is any hyperplane. Therefore

(D1 · · · · · Dn) = δ1 · · · δn

where δj is the degree of Dj . For example, if C1 and C2 are curves in P2 defined by

irreducible polynomials F1 and F2 of degrees δ1 and δ2 respectively, then C1 and C2

intersect in δ1δ2 points (counting multiplicities).

References:

Shafarevich 1994, IV.1, IV.2.

Fulton, W., Introduction to Intersection Theory in Algebraic Geometry, (AMS

Publication; CBMS regional conference series #54.) This is a pleasant introduction.

Fulton, W., Intersection Theory. Springer, 1984. The ultimate source for everything

to do with intersection theory.

Serre:A lg`ebre Locale, Multiplicit´es, Springer Lecture Notes, 11, 1957/58 (third

edition 1975). This is where the definition in terms of Tor’s was first suggested.

Algebraic Geometry: 11. Coherent Sheaves; Invertible Sheaves 143

Навигация