Line bundles and maps to projective space

Paul Hacking

3/23/10

Let X be a compact complex manifold and p : L → X a holomorphicline bundle.

Let s0, . . . , sN be a basis of the space Γ(X,L) of global sections of L.(Note: If X is compact then the complex vector space Γ(X,L) is finitedimensional. This is a hard theorem.)

For each P ∈ X, si(P ) is an element of the fiber LP of L over P ∈ X, a1-dimensional complex vector space. So, provided si(P ) 6= 0 for some i, weget a point

(s0(P ) : · · · : sN (P )) ∈ PN

To be precise, we choose a basis {v} of LP and write si(P ) = λi · v whereλi ∈ C, then

(s0(P ) : · · · : sN (P )) := (λ0 : · · · : λN ).

This does not depend on the choice of v.Letting P ∈ X vary, we get a rational map

φ : X 99K PN , P 7→ (s0(P ) : · · · : sN (P )).

More precisely, φ is a morphism

φ : U → PN

of complex manifolds with domain containing the dense open set U ⊂ Xgiven by

U = X \ (s1 = · · · = sN = 0).

In particular, φ is a morphism if (s1 = · · · = sN = 0) is empty, equivalently,for all P ∈ X there exists a global section s of L such that s(P ) 6= 0. In thiscase we say L is globally generated.

Essentially all maps to projective space are of this form: Given a mor-phism

φ : X → PN ,

1

let L := φ∗H be the pullback of the line bundle H on PN (the dual of thetautological line bundle). The homogeneous coordinates X0, . . . , XN on PN

give a basis of global sections of H. Let si := φ∗Xi be the correspondingsections of L. Then φ = (s0 : · · · : sN ) as above. Note: In general, it is nottrue that s0, . . . , sN is a basis of Γ(X,L). But this is a minor point: we cancomplete s0, . . . , sN to a basis s0, . . . , sM , with N ≤ M , this defines a mapφ̃ : X → PM , and φ is the composition of φ̃ with the projection

PM 99K PN , (X0 : · · · : XM ) 7→ (X0 : · · · : XN ).

Notice that the zero loci of global sections of L are the inverse imagesof hyperplanes in PN under the map φ (counted with multiplicities): givens ∈ Γ(X,L) write s =

∑λisi = φ∗(

∑λiXi), then

(s = 0) = φ−1(∑

λiXi = 0).

Example 0.1. Let X = P1, L = H⊗2. Then Γ(X,L) is the space of homoge-neous polynomials of degree 2 in the homogeneous coordinates Y0, Y1 on X.Then L and the basis s0 = Y 2

0 , s1 = Y0Y1, s2 = Y 21 defines a morphism

φ : X → P2, (Y0 : Y1) 7→ (X0 : X1 : X2) = (Y 20 : Y0Y1 : Y 2

1 ).

This is an embedding with image the conic

φ(X) = (X0X2 = X21 ) ⊂ P2.

Example 0.2. Let X = C/L be an elliptic curve (a complex torus of dimen-sion 1). Here L = Zl1 + Zl2 ⊂ C is a lattice generated by two complexnumbers l1, l2 ∈ C which are linearly independent over R. So as a real man-ifold X is isomorphic to the 2-torus R2/Z2 = S1 × S1, but as a complexmanifold the isomorphism type of X depends on the ratio l2/l1.

Let P ∈ X be the origin of the group law on X (the image of 0 ∈ C1z). Let

L = O(2P ) be the line bundle on X associated to the divisor D = 2P . Recallthat in general for a Riemann surface surface X and a divisor D =

∑ri=1 niPi

on X we can identify the space of global sections of O(D) with the spaceof meromorphic functions f on X which are holomorphic away from the Pi

and satisfyordPi(f) ≥ −ni

for each i (where ordP (f) ∈ Z denotes the order of vanishing of a meromor-phic function f at a point P ∈ X). This is given by the trivialization of theline bundle O(D) over the open set U = X \ {P1, . . . , Pr}. In the present

2

case we find Γ(X,L) is identified with space of meromorphic functions onX having a pole of order at most 2 at P and holomorphic elsewhere. Thisspace has basis 1, ℘ given by the constant function 1 and the Weierstrass ℘function

℘(z) =1z2

+∑

l∈L\{0}

(1

(z − l)2− 1l2

).

(This function is a doubly periodic meromorphic function on the universalcover C of X with coordinate z which induces a meromorphic function onthe quotient X = C/L. It has a pole of order 2 at P ∈ X and is holomorphicelsewhere.) Let s0, s1 be the basis of Γ(X,L) corresponding to 1, ℘. ThenL and s0, s1 define a morphism

φ : X → P1, φ = (s0 : s1) = (1 : ℘).

The zero locus of s0 (as a section of L) is 2P . (In general the zero locus ofthe section of O(D) corresponding to the constant function 1 is D.) Now(s0 = 0) is the inverse image of the hyperplane (X0 = 0) (which in this caseis the point (0 : 1) ∈ P1) under the map φ, so we see that φ has degree 2and is ramified at P . According to Riemann–Huritz there are 4 ramificationpoints in total. Noting that ℘ is an even function of z we see that φ is thequotient map for the involution of X given by z 7→ −z and the ramificationpoints are given by the points 1

2L/L of order 2 on X = C/L.Note that if we take L = O(P ) then Γ(X,L) is 1-dimensional and so we

do not get anything interesting (φ : X → P0 is the map to a point).Now let L = O(3P ). Then the space of global sections of L is identified

with the meromorphic functions on X having a pole of order at most 3at P and holomorphic elsewhere, which has basis 1, ℘, ℘′ (where the primedenotes the derivative with respect to z). The corresponding basis s0, s1, s2of Γ(X,L) defines an embedding

φ : X → P2.

(We will omit the proof that φ is an embedding, it is a good exercise.)The zero locus of s0 is 3P . This is the inverse image of the hyperplane(X0 = 0) ⊂ P2. So φ(X) is a curve of degree 3 in P2 (because the line(X0 = 0) intersects φ(X) in 3 points (counting with multiplicities)) andφ(P ) is a flex (or inflection point) of the plane curve φ(X), that is, thetangent line to the curve at φ(P ) has order of contact equal to 3 (becauseφ−1(X0 = 0) = P counted with multiplicity 3).

3

In general, given a compact complex manifold X and a holomorphicline bundle p : L → X, we’d like to know when the induced rational mapφ : X 99K PN is (a) a morphism and (b) an embedding. We have alreadyseen that φ is a morphism iff for all P ∈ X there exists s ∈ Γ(X,L) suchthat s(P ) 6= 0. Similarly, φ is an embedding iff in addition the followingconditions are satisfied:

(1) For all pairs of distinct points P,Q ∈ X there exists s ∈ Γ(X,L) suchthat s(P ) 6= 0 and s(Q) = 0.

(2) For all P ∈ X there exists s1, . . . , sn ∈ Γ(X,L) such that si(P ) = 0for each i and det( ∂si

∂zj(P )) 6= 0, where n is the dimension of X and

z1, . . . , zn are local coordinates at P .

Here (1) implies φ is injective as a map of sets and (2) implies φ is a localisomorphism at P by the holomorphic version of the inverse function theo-rem. In the case n = 1 condition (2) can be stated more simply as follows:for all P ∈ X there exists s ∈ Γ(X,L) such that the order of vanishing of sat P equals 1.

These problems are typical of algebraic geometry in general in that weare looking for global holomorphic objects (in this case, global sections of aline bundle) satisfying local conditions. To answer these questions we needthe theory of cohomology of sheaves. Please see [GH78, p. 34–43] for anintroduction to this topic.

References

[GH78] P. Griffiths, J. Harris, Principles of algebraic geometry, Wiley 1978.

4