IfX is a subscheme ofPNk of dimensionn, it is proved in [H1], Theorem I.7.5, that the function
m7→χ(X,OX(m))
ispolynomial of degreen, i.e., takes the same values on the integers as a (uniquely determined) polynomial of degreenwith rational coefficients, called theHilbert polynomial of X. The degree of X in PNk is then defined as n! times the co-efficient ofmn. It generalizes the degree of a hypersurface defined in Example 2.12.
IfX is reduced andH1, . . . , Hn are general hyperplanes, the degree ofX is also the number of points of the intersection X∩H1∩ · · · ∩Hn. If HiX is the Cartier divisor onX defined byHi, the degree ofX is therefore the number of points in the intersectionH1X∩· · ·∩HnX. Our aim in this section is to generalize this and to define an intersection number
(D1·. . .·Dn)
for any Cartier divisorsD1, . . . , Dnon a projectiven-dimensional scheme, which only depends on the linear equivalence class of theDi.
Instead of trying to define, as in Definition 3.5, the multiplicity of intersection at a point, which can be difficult on a general X, we give a definition based on Euler characteristics, as in Theorem 3.6 (compare with (3.3)). It has the advantage of being quick and efficient, but has very little geometric feeling to it.
Theorem 3.12 Let D1, . . . , Dr be Cartier divisors on a projective scheme X. The function
(m1, . . . , mr)7−→χ(X, m1D1+· · ·+mrDr)
takes the same values onZr as a polynomial with rational coefficients of degree at most the dimension ofX.
Proof. We prove the theorem first in the case r = 1 by induction on the dimension ofX. IfX has dimension 0, we have
χ(X, D) =h0(X,OX) for anyD and the conclusion holds trivially.
Write D1 = D ∼lin E1−E2 with E1 and E2 effective (Proposition 2.36).
There are exact sequences
0→ OX(mD−E1) → OX(mD) → OE1(mD) →0
0→ OX((m−k1)D−E2) → OX((m−1)D) → OE2((m−1)D) →0 (3.2) which yield
χ(X, mD)−χ(X,(m−1)D) =χ(E1, mD)−χ(E2,(m−1)D).
By induction, the right-hand side of this equality is a rational polynomial func-tion in m of degree d < dim(X). But if a function f : Z → Z is such that m7→f(m)−f(m−1) is rational polynomial of degree δ, the function f itself is rational polynomial of degree δ+ 1 ([H1], Proposition I.7.3.(b)); therefore, χ(X, mD) is a rational polynomial function inmof degree≤d+ 1≤dim(X).
Note that for any divisorD0 on X, the function m7→χ(X, D0+mD) is a rational polynomial function of degree≤dim(X) (the same proof applies upon tensoring the diagram (3.2) byOX(D0)). We now treat the general case.
Lemma 3.13 Letdbe a positive integer and letf :Zr→Zbe a map such that for each(n1, . . . , ni−1, ni+1, . . . , nr)inZr−1, the map
m7−→f(n1, . . . , ni−1, m, ni+1, . . . , nr)
is rational polynomial of degree at mostd. The functionf takes the same values as a rational polynomial inrindeterminates.
Proof. We proceed by induction on r, the case r = 1 being done in [H1], Proposition I.7.3.(a). Assumer >1; there exist functionsf0, . . . , fd:Zr−1→Z such that
f(m1, . . . , mr) =
d
X
j=0
fj(m1, . . . , mr−1)mjr
Pick distinct integers c0, . . . , cd; for each i ∈ {0, . . . , d}, there exists by the induction hypothesis a polynomialPi with rational coefficients such that
f(m1, . . . , mr−1, ci) =
d
X
j=0
fj(m1, . . . , mr−1)cji =Pi(m1, . . . , mr−1) The matrix (cji) is invertible and its inverse has rational coefficients. This proves that eachfjis a linear combination ofP0, . . . , Pdwith rational coefficients hence
the lemma.
From the remark before Lemma 3.13 and the lemma itself, we deduce that there exists a polynomialP ∈Q[T1, . . . , Tr] such that
χ(X, m1D1+· · ·+mrDr) =P(m1, . . . , mr)
for all integers m1, . . . , mr. Let d be its total degree, and let n1, . . . , nr be integers such that the degree of the polynomial
Q(T) =P(n1T, . . . , nrT) is stilld. Since
Q(m) =χ(X, m(n1D1+· · ·+nrDr))
it follows from the caser= 1 thatdis at most the dimension ofX.
Definition 3.14 Let D1, . . . , Dr be Cartier divisors on a projective scheme X withr≥dim(X). We define the intersection number
(D1·. . .·Dr)
as the coefficient ofm1· · ·mr in the rational polynomial χ(X, m1D1+· · ·+mrDr).
Of course, this number only depends on the linear equivalence classes of the divisorsDi, since it is defined from the invertible sheaves OX(Di). In case X is a subscheme ofPNk, and ifHX is a hyperplane section ofX, the intersection number ((HX)n) is the degree ofX as defined in [H1], §I.7. vanishes for all other monomials of degree≤r). It follows that we have
(D1·. . .·Dr) = X
I⊂{1,...,r}
εI χ(X,−X
i∈I
Di). (3.3)
This number is therefore an integer and it vanishes forr >dim(X) (Theorem 3.12). IfD1, . . . , Drare effective and meet properly in a finite number of points, the intersection number does have a geometric interpretation as the number of points inD1∩ · · · ∩Dr, counted with multiplicity (see [Ko1], Theorem VI.2.8;
when the intersection is transverse, the proof that
(D1·. . .·Dr) = Card(D1∩ · · · ∩Dr) is the same as that of Theorem 3.6).
Of course, it coincides with our previous definition on surfaces (compare (3.3) with (3.1)). On a curveX, we can use it to define the degree of a Cartier divisorDby setting deg(D) = (D). The Riemann-Roch theorem then becomes a tautology. Given a morphismf :C→X from a projective curve to a projective schemeX, and a Cartier divisorD onX, we define
(D·C) = deg(f∗D). (3.4) Finally, ifD is a Cartier divisor on the projectiven-dimensional scheme X, the functionm7→χ(X, mD) is a polynomialP(T) =Pn
The coefficient ofm1· · ·mn in this polynomial isann!, hence χ(X, mD) =mn(Dn)
n! +O(mn−1). (3.5)
We now prove multilinearity.
Proposition 3.15 Let D1, . . . , Dn be Cartier divisors on a projective scheme X of dimensionn.
a) The map
(D1, . . . , Dn)7−→(D1·. . .·Dn) isZ-multilinear, symmetric and takes integral values.
b) IfDn is effective,
(D1·. . .·Dn) = (D1|Dn·. . .·Dn−1|Dn).
Proof. The map in a) is symmetric by construction, but its multilinearity is not obvious. The right-hand side of (3.3) vanishes forr > n, hence, for any divisorsD1, D10, D2, . . . , Dn, the sum Putting all these identities together gives the desired equality
((D1+D01)·D2·. . .·Dn) = (D1·D2·. . .·Dn) + (D10 ·D2·. . .·Dn) and proves a).
In the situation of b), we have (D1·. . .·Dn) = X
From the exact sequence
Using multilinearity, we see that the degree of a divisor on a curve X that we just defined agrees with the previous definition (for smooth curves over an algebraically closed field): for one smooth pointpofX, the exact sequence
0→OX→OX(p)→k(p)→0 (3.6)
yields deg(p) = 1.
Recall that the degree of a dominant morphismπ:Y →X between varieties is the degree of the field extensionπ∗:K(X),→K(Y) if this extension is finite, and 0 otherwise.
Proposition 3.16 (Pull-back formula) Letπ:Y →X be a surjective mor-phism between projective varieties. LetD1, . . . , Drbe Cartier divisors onX with r≥dim(Y). We have
(π∗D1·. . .·π∗Dr) = deg(π)(D1·. . .·Dr).
Sketch of proof. For any coherent sheaf F onY, the sheaves Rqπ∗F are coherent ([G1], th. 3.2.1) and there is a spectral sequence
Hp(X, Rqπ∗F) =⇒Hp+q(Y,F).
(Here we need an extension of Theorem 3.12 which says that for any coherent sheafF onX, the function
(m1, . . . , mr)7−→χ(X,F(m1D1+· · ·+mrDr)) is still polynomial of degree≤dim(SuppF).)
If π is not generically finite, we have r > dim(X) and the coefficient of m1· · ·mr in each term of the sum vanishes by Theorem 3.12.
Otherwise,πis finite of degreedover a dense open subsetUofY, the sheaves Rqπ∗OY have support outside ofU forq >0 ([H1], Corollary III.11.2) hence the coefficient ofm1· · ·mrin the corresponding term vanishes for the same reason.
Finally,π∗OY is free of rankdon some dense open subset ofU and it is not too hard to conclude that the coefficients ofm1· · ·mrinχ(X, π∗OY ⊗OX(m1D1+
· · ·+mrDr)) andχ(X,OX⊕d⊗OX(m1D1+· · ·+mrDr)) are the same.
3.17. Projection formula. Letπ:X →Y be a morphism between projective varieties and letC be a curve onX. We define the 1-cycle π∗C as follows: if C is contracted to a point by π, set π∗C = 0; if π(C) is a curve on Y, set π∗C =d π(C), where d is the degree of the morphismC → π(C) induced by π. If D is a Cartier divisor on Y, we obtain from the pull-back formula the so-calledprojection formula
(π∗D·C) = (D·π∗C). (3.7) Corollary 3.18 LetX be a curve of genus0over a fieldk. IfX has ak-point, X is isomorphic toP1k.
Any plane conic with no rational point (such as the real conic with equation x20+x21+x22= 0) has genus 0 (see Exercise 3.2), but is of course not isomorphic to the projective line.
Proof. Let p be a k-point of X. Since H1(X,OX) = 0, the long exact sequence in cohomology associated with (3.6) gives an exact sequence
0→H0(X,OX)→H0(X,OX(p))→kp→0.
In particular, h0(X,OX(p)) = 2 and the invertible sheaf OX(p) is generated by two global sections which define a finite morphismu : X → P1k such that u∗OP1k(1) =OX(p). By the pull-back formula,
1 = deg(OX(p)) = deg(u),
anduis an isomorphism.
Exercise 3.19 Let E be the exceptional divisor of the blow-up of a smooth point on ann-dimensional projective scheme (see§3.3.2). Compute (En).
3.20. Intersection of Q-divisors. Of course, we may define, by linearity, intersection of Q-Cartier Q-divisors. For example, let X be the cone in P3k with equationx0x1=x22 (its vertex is (0,0,0,1)) and letL be the line defined byx0=x2= 0 (compare with Example 2.6). Then 2Lis a hyperplane section ofX, hence (2L)2= deg(X) = 2. So we have (L2) = 1/2.