Proposition 2.36 Any Cartier divisor on a projective scheme is linearly equiv-alent to the difference of two effective Cartier divisors.
Proof. Assume for simplicity that the projective scheme X is integral. Let D be a Cartier divisor on X and letH be an effective very ample divisor on X. For m 0, the invertible sheaf OX(D+mH) is generated by its global sections. In particular, it has a nonzero section; letE be its (effective) divisor.
We have
D∼lin E−mH,
which proves the proposition.
2.8 A cohomological characterization of ample divisors
Theorem 2.37 LetX be a projective scheme over a field and letDbe a Cartier divisor onX. The following properties are equivalent:
(i) D est ample;
(ii) for each coherent sheaf F on X, we have Hq(X,F(mD)) = 0 for all m0 and all q >0;
(iii) for each coherent sheaf F on X, we have H1(X,F(mD)) = 0 for all m0.
Proof. Assume D ample. Theorem 2.34 then implies thatrD is very ample for somer >0. For each 0≤s < r, Corollary 2.29.b) yields
Hq(X,(F(sD))(mD)) = 0 for allm≥ms. For
m≥rmax(m0, . . . , mr−1),
we haveHq(X,F(mD)) = 0. This proves that (i) implies (ii), which trivially implies (iii).
Assume that (iii) holds. Let F be a coherent sheaf on X, letxbe a closed point ofX, and letG be the kernel of the surjection
F →F⊗k(x)
ofOX-modules. Since (iii) holds, there exists an integerm0such that H1(X,G(mD)) = 0
for allm≥m0 (note that the integerm0may depend on F and x). Since the sequence
0→G(mD)→F(mD)→F(mD)⊗k(x)→0 is exact, the evaluation
Γ(X,F(mD))→Γ(X,F(mD)⊗k(x))
is surjective. This means that its global sections generateF(mD) in a neigh-borhoodUF,m ofx. In particular, there exists an integerm1 such thatm1Dis globally generated onUOX,m1. For all m≥m0, the sheaf F(mD) is globally generated on
Ux=UOX,m1∩UF,m0∩UF,m0+1∩ · · · ∩UF,m0+m1−1
since it can be written as
(F((m0+s)D))⊗OX(r(m1D))
withr≥0 and 0≤s < m1. CoverX with a finite number of open subsetsUx
and take the largest corresponding integerm0. This shows thatDis ample and
finishes the proof of the theorem.
Corollary 2.38 Let X and Y be projective schemes over a field and let u : X →Y be a morphism with finite fibers. Let D be an ampleQ-Cartier divisor onY. Then theQ-Cartier divisoru∗D is ample.
Proof. We may assume that D Cartier divisor. Let F be a coherent sheaf on X. In our situation, the sheaf u∗F is coherent ([H1], Corollary II.5.20).
Moreover, the morphism uis finite5 and the inverse image by uof any affine open subset of Y is an affine open subset of X ([H1], Exercise II.5.17.(b)). If U is a covering ofY by affine open subsets,u−1(U) is then a covering ofX by affine open subsets, and by definition ofu∗F, the associated cochain complexes are isomorphic. This implies
Hq(X,F)'Hq(Y, u∗F) for all integersq. We now have (projection formula)
u∗(F(mu∗D))'(u∗F)(mD) hence
H1(X,F(mu∗D))'H1(Y,(u∗F)(mD)).
Sinceu∗Fis coherent andDis ample, the right-hand-side vanishes for allm0 by Theorem 2.37, hence also the left-hand-side. By the same theorem, it follows
thatthe divisoru∗D est ample.
Exercise 2.39 In the situation of the corollary, ifuisnotfinite, show thatu∗D isnotample.
Exercise 2.40 LetX be a projective scheme over a field. Show that a Cartier divisor is ample onX if and only if it is ample on each irreducible component ofXred.
5The very important fact that a projective morphism with finite fibers is finite is deduced in [H1] from the difficult Zariski’s Main Theorem. In our case, it can also be proved in an elementary fashion (see [D2], th. 3.28).
Chapter 3
Intersection of curves and divisors
3.1 Curves
A curve is a projective integral scheme X of dimension 1 over a field k. We define its (arithmetic) genus as
g(X) = dimH1(X,OX).
Example 3.1 The curveP1khas genus 0. This can be obtained by a computa-tion in Cech cohomology: coverX with the two affine subsetsU0andU1. The Cech complex
Γ(U0,OU0)⊕Γ(U1,OU1)→Γ(U01,OU01) is
k[t]⊕k[t−1]→k[t, t−1], hence the result.
Exercise 3.2 Show that the genus of a plane curve of degreedis (d−1)(d−2)/2 (Hint: assume that (0,0,1) is not on the curve, cover it with the affine subsets U0 andU1and compute the Cech cohomology groups as above).
We defined in Example 2.7 the degree of a Cartier divisor (or of an invertible sheaf) on a smooth curve over a fieldkby setting
deg X
pclosed point inX
npp
=X
np[k(p) :k].
In particular, whenk is algebraically closed, this is justP np.
IfD =P
pnppis an effective divisor (np ≥0 for allp), we can view it as a 0-dimensional subscheme ofX with (affine) support at set of pointspfor which np>0, where it is defined by the ideal mnX,pp . We have
h0(D,OD) =X
p
dimk(OX,p/mnX,pp ) =X
p
npdimk(OX,p/mX,p) = deg(D).
The central theorem in this section is the following.1
Theorem 3.3 (Riemann-Roch theorem) Let X be a smooth curve. For any divisorD onX, we have
χ(X, D) = deg(D) +χ(X,OX) = deg(D) + 1−g(X).
Proof.By Proposition 2.36, we can writeD ∼lin E−F, whereE andF are effective (Cartier) divisors on X. Considering them as (0-dimensional) sub-schemes ofX, we have exact sequences (see Remark 2.10)
0→ OX(E−F) → OX(E) → OF →0
0→ OX → OX(E) → OE →0
(note that the sheafOF(E) is isomorphic toOF, becauseOX(E) is isomorphic to OXin a neighborhood of the (finite) support ofF, and similarly,OE(E)'OE).
As remarked above, we have
χ(F,OF) =h0(F,OF) = deg(F).
Similarly,χ(E,OE) = deg(E). This implies χ(X, D) = χ(X, E)−χ(F,OF)
= χ(X,OX) +χ(E,OE)−deg(F)
= χ(X,OX) + deg(E)−deg(F)
= χ(X,OX) + deg(D),
and the theorem is proved.
Later on, we will use this theorem todefinethe degree of a Cartier divisorD on any curveX, as the leading term of (what we will prove to be) the degree-1 polynomialχ(X, mD). The Riemann-Roch theorem then becomes a tautology.
Corollary 3.4 Let X be a smooth curve. A divisor D on X is ample if and only ifdeg(D)>0.
1This should really be called the Hirzebruch-Riemann-Roch theorem (or a (very) particular case of it). The original Riemann-Roch theorem is our Theorem 3.3 with the dimension of H1(X,L) replaced with that of its Serre-dualH0(X, ωX⊗L−1).
This will be generalized later to any curve (see 4.2).
Proof. Let p be a closed point of X. If D is ample, mD−p is linearly equivalent to an effective divisor for somem0, in which case
0≤deg(mD−p) =mdeg(D)−deg(p), hence deg(D)>0.
Conversely, assume deg(D)>0. By Riemann-Roch, we haveH0(X, mD)6= 0 form0, so, upon replacingD by a positive multiple, we can assume that Dis effective. As in the proof of the theorem, we then have an exact sequence
0→OX((m−1)D)→OX(mD)→OD→0, from which we get a surjection2
H1(X,(m−1)D))→H1(X, mD)→0.
Since these spaces are finite-dimensional, this will be a bijection form0, in which case we get a surjection
H0(X, mD)→H0(D,OD).
In particular, the evaluation map evx (see §2.5) for the sheaf OX(mD) is sur-jective at every pointxof the support of D. Since it is trivially surjective forx outside of this support (it has a section with divisormD), the sheafOX(mD) is globally generated.
Its global sections therefore define a morphism u : X → PNk such that OX(mD) =u∗OPNk(1). Since OX(mD) is non trivial,uis not constant, hence finite becauseX is a curve. But then,OX(mD) =u∗OPNk(1) is ample (Corollary
2.38) henceDis ample.