3 Line bundles and divisors

General references:  [ GW1 ] Ch. 11, in particular (11.9), (11.13); [ H ]  II.6.

A divisor on a scheme \(X\) should be thought of an object that encodes a “configuration of zeros and poles (with multiplicities)” that a function on \(X\) could have. Below, we will see two ways to make this precise and compare them.

Let \(X\) be an integral (i.e., reduced and irreducible) scheme. We denote by \(K(X)\) the field of rational functions of \(X\).

Later we will impose the additional condition that \(X\) is noetherian and that all local rings \({\mathscr O}_{X,x}\) are unique factorization domains.

An important example that is good to keep in mind is the case of a Dedekind scheme of dimension \(1\), i.e., \(X\) is a noetherian integral scheme such that all points except for the generic point are closed, and such that for every closed point \(x\in X\) the local ring \({\mathscr O}_{X, x}\) is a principal ideal domain (in other words: all local rings are discrete valuation rings), and the generic point is not closed itself. If a Dedekind scheme \(X\) is a \(k\)-scheme of finite type for some algebraically closed (or at least perfect) field \(k\), then we call \(X\) a smooth algebraic curve over \(k\).

Cartier divisors


(3.1) Cartier divisors: Definition

Denote by \(K(X) ={\mathscr O}_{X, \eta }\) the field of rational functions on the integral scheme \(X\), where \(\eta \in X\) is the generic point. We denote by \({\mathscr K}_X\) the constant sheaf with value \(K(X)\), i.e., \({\mathscr K}_X(U) = K(X)\) for all \(\emptyset \ne U\subseteq X\) open. Since \(X\) is irreducible, this is a sheaf.

The notion of Cartier divisor encodes a zero/pole configuration by specifying, locally on \(X\), functions with the desired zeros and poles. Since functions which are units in \(\Gamma (U, {\mathscr O}_X)\) should be regarded as having no zeros and/or poles on \(U\), we consider functions only up to units.

Definition 3.1
A Cartier divisor on \(X\) is given by a tuple \((U_i, f_i)_i\), where \(X=\bigcup _i U_i\) is an open cover, \(f_i \in K(X)^\times \), and \(f_i/f_j \in \Gamma (U_i\cap U_J, {\mathscr O}_X)^\times \) for all \(i, j\). Two such tuples \((U_i, f_i)_i\), \((Vj, g_j)_j\) give rise to the same divisor, if \(f_ig_j^{-1}\in \Gamma (U_i\cap V_j, {\mathscr O}_X)^\times \) for all \(i\), \(j\).

May 15,

With addition given by

\[ (U_i, f_i)_i + (V_j, g_j)_j = (U_i\cap V_j, f_ig_j)_{i,j} \]

the set \(\operatorname{Div}(X)\) of all Cartier divisors on \(X\) is an abelian group.

Remark 3.2
We have \(\operatorname{Div}(X) = \Gamma (X, {\mathscr K}_X^\times /{\mathscr O}_X^\times )\).

Definition 3.3
A Cartier divisor of the form \((X, f)\), \(f\in K(X)^\times \), is called a principal divisor. Divisors \(D\), \(D'\) on \(X\) are called linearly equivalent, if \(D-D'\) is a principal divisor. The set of principal divisors is a subgroup of \(\operatorname{Div}(X)\) and the quotient \(\operatorname{DivCl}(X)\) of \(\operatorname{Div}(X)\) by this subgroup is called the divisor class group of \(X\).

(3.2) The line bundle attached to a Cartier divisor

Let \(D\) be a Cartier divisor on \(X\). We define an invertible \({\mathscr O}_X\)-module \({\mathscr O}_X(D)\) as follows:

\[ \Gamma (U, {\mathscr O}_X(D)) = \{ f\in K(X); \forall i: f_if\in \Gamma (U\cap U_i, {\mathscr O}_X) \} \quad \text{for}\ \emptyset \ne U\subseteq X\text{\ open.} \]

For each \(i\), we have \({\mathscr O}_X(D)_{|U_i} = f_i^{-1}{\mathscr O}_{U_i}\subset {\mathscr K}_X\), so multiplication by \(f_i\) gives an \({\mathscr O}_{U_i}\)-module isomorphism \({\mathscr O}_X(D)_{|U_i}\cong {\mathscr O}_{U_i}\).

Proposition 3.4
The map \(D\mapsto {\mathscr O}_D(X)\) induces group isomorphisms \(\operatorname{Div}(X) \cong \{ {\mathscr L}\subset {\mathscr K}_X\ \text{invertible}\ {\mathscr O}_X\text{-module} \} \) and \(\operatorname{DivCl}(X) \cong \operatorname{Pic}(X)\).

Sketch of proof

To construct an inverse of the map \(D\mapsto {\mathscr O}_D(X)\), take \({\mathscr L}\subseteq {\mathscr K}_X\) invertible and choose an open cover \(X=\bigcup U_i\) such that \({\mathscr L}_{|U_i}\) is trivial for each \(i\). Then necessarily \({\mathscr L}_{|U_i} = f_i^{-1}{\mathscr O}_{U_i}\) for some \(f_i\in K(X)^\times \) (namely, \(f_i^{-1}\) is the image of \(1\in \Gamma (U_i,{\mathscr O}_X)\) under the map \(\Gamma (U_i,{\mathscr O}_X)\to K(X)\) induced by the composition \({\mathscr O}_{U_i}\cong {\mathscr L}_{|U_i}\to {\mathscr K}_X\)). We then map \({\mathscr L}\) to the Cartier divisor \((U_i, f_i)_i\). One checks that this map is well-defined (i.e., independent of the choice of cover and of the choice of the elements \(f_i\)) and that the two maps are inverse to each other.

It remains to check that \({\mathscr O}_D(X)\) is free if and only if \(D\) is principal (Problem 28) and that every invertible \({\mathscr O}_X\)-module \({\mathscr L}\) can be embedded as a submodule into \({\mathscr K}_X\). This is easy if \(X\) is affine. In the general case, let \(U\subseteq X\) be open affine. We claim that every embedding \({\mathscr L}_{|U}\hookrightarrow {\mathscr K}_U\) extends uniquely to an embedding \({\mathscr L}\hookrightarrow {\mathscr K}_X\). Because of the uniqueness, we can work locally on \(X\) (and afterwards use gluing of sheaf homomorphisms), and therefore restrict to the case \({\mathscr L}= {\mathscr O}_X\). The embedding \({\mathscr O}_U={\mathscr L}_{|U}\hookrightarrow {\mathscr K}_U\) then corresponds to a section \(s\in \Gamma (U, {\mathscr K}_U)^\times \). But \(\Gamma (U, {\mathscr K}_U)=K(X) =\Gamma (X, {\mathscr K}_X)\), so the claim follows. (See  [ GW1 ] Prop. 11.29 for more details and a variant which does not require \(X\) to be integral.)

To get a more geometric view on divisors, a first step is the following definition of the support of a divisor. We will carry this further by introducing the notion of Weil divisor, see below, and relating it to Cartier divisors.

Definition 3.5
The support of a Cartier divisor \(D\) is
\[ \operatorname{Supp}(D) = \{ x\in X;\ f_{i, x} \in K(X)^\times \setminus {\mathscr O}_{X,x}^\times \ \text{(where}\ x\in U_i\text{)}\} , \]
a proper closed subset of \(X\).

Weil divisors


May 17,

Now let \(X\) be a noetherian integral scheme, such that all local rings \({\mathscr O}_{X,x}\) are unique factorization domains. (The theory can be set up in more generality, see  [ GW1 ] Section (11.13).) This is true for example if all local rings of \(X\) are regular local rings.

(Recall that a noetherian local ring \(A\) is called regular, if its maximal ideal can be generated by \(\dim A\) elements, where \(\dim A\) is the Krull dimension of \(A\). The Krull dimension is the supremum of the lengths of inclusion chains of prime ideals in \(A\). For instance, a local ring \(A\) has dimension \(0\) if and only if its maximal ideal is also a minimal prime ideal (and hence the unique prime ideal of \(A\)). A local ring \(A\) has dimension \(1\) if it maximal ideal is not a minimal prime ideal, but all non-maximal prime ideals are minimal prime ideals. A local domain has dimension \(1\) if and only if it has precisely two prime ideals.)

(3.3) Definition of Weil divisors

Let \(Z^1(X)\) denote the free abelian group on maximal proper integral subschemes of \(X\) (equivalently: those integral subschemes \(Z\subset X\) such that for the generic point \(\eta _Z\in Z\) we have \(\dim {\mathscr O}_{X, \eta _Z} = 1\)). We say that \(Z\) has codimension \(1\). We also write \({\mathscr O}_{X,Z}:={\mathscr O}_{X, \eta _Z}\).

By our assumptions on \(X\), all the rings \({\mathscr O}_{X, Z}\) are discrete valuation rings. (Since they are noetherian domains of dimension \(1\) by assumption, it is equivalent to require that they are integrally closed, or factorial, or that they are regular.) We denote by \(v_Z\colon K(X)^\times \to \mathbb {Z}\) the corresponding discrete valuation on \(K\), and set \(v_Z(0)=\infty \).

Definition 3.6
An element of \(Z^1(X)\) is called a Weil divisor. We write Weil divisors as finite “formal sums” \(\sum n_Z[Z]\) where \(Z\subset X\) runs through the integral closed subschemes of \(X\) of codimension \(1\).

For \(f\in K(X)^\times \), we define the divisor attached to \(f\) as

\[ \mathop{\rm div}\nolimits (f) = \sum _Z v_Z(f) [Z]. \]

Note that the sum is finite, i.e., \(v_Z(f)=0\) for all but finitely many \(Z\). In fact, for \(U\subseteq X\) affine open, the complement \(X\setminus U\) has only finitely many irreducible components, so we may discard it and replace \(X\) by \(U\). Then assume \(X=\operatorname{Spec}A\) is affine and write \(f = g/h\) with \(g,h\in A\). Then \(v_Z(f)\) can only be \(\ne 0\), if \(Z\) is an irreducible component of \(V(f)\cup V(g)\). Since this closed subscheme of the noetherian scheme \(X\) has only finitely many irreducible components (being itself noetherian), we are done.

Weil divisors of the form \(\mathop{\rm div}\nolimits (f)\) are called principal Weil divisors. Two Weil divisors are called linearly equivalent, if their difference is a principal divisor.

Definition 3.7
  1. A Weil divisor \(\sum _Z n_Z[Z]\) is called effective, if \(n_Z\ge 0\) for all \(Z\).

  2. A Cartier divisor \(D\) is called effective, if \({\mathscr O}_X\subseteq {\mathscr O}_X(D)\) (inside \({\mathscr K}_X\)), or equivalently, if \({\mathscr O}_X(-D)\subseteq {\mathscr O}_X\) is an ideal of \({\mathscr O}_X\).

(3.4) Weil divisors vs. Cartier divisors

Generalizing the definition of principal divisors, we can construct a group homomorphism \(\operatorname{cyc}\colon \operatorname{Div}(X)\to Z^1(X)\) as follows:

\[ D = (U_i, f_i) \mapsto \sum v_Z(f_{i_Z}) [Z], \]

where for each \(Z\) we choose an index \(i_Z\) so that \(U_{i_Z}\) contains the generic point of \(Z\) (equivalently: \(U_{i_Z}\cap Z\ne \emptyset \)).

To prove that \(\operatorname{cyc}\) is an isomorphism \(\operatorname{Div}(X)\cong Z^1(X)\), we need the following facts from commutative algebra. (See, e.g., [ M2 ] Theorem 11.5, Theorem 20.1.) Recall that a domain \(A\) is called integrally closed, or normal, if every element of the field of fractions of \(A\) which is the zero of a monic polynomial with coefficients in \(A\) lies in \(A\). Every UFD is integrally closed. Furthermore, a domain \(A\) is integrally closed if and only if all localizations \(A_{\mathfrak p}\) at prime ideals \(\mathfrak p\in \operatorname{Spec}A\) are integrally closed. So for \(X\) as above, and a non-empty affine open \(U\subseteq X\), the ring \(\Gamma (U, {\mathscr O}_X)\) is integrally closed (in its field of fractions \(K(X)\)).

Lemma 3.8
  1. Let \(A\) be an integrally closed domain. Then \(A\) is equal to the intersection of all localizations \(A_{\mathfrak p}\) (in \(\operatorname{Frac}(A)\)), where \(\mathfrak p\) runs through the set of minimal prime ideals of \(A\).

  2. Let \(A\) be a local unique factorization domain. Then every prime ideal \(\mathfrak p\ne 0\) which is minimal among all prime ideals \(\ne 0\) of \(A\) is a principal ideal.

Proposition 3.9
The map \(\operatorname{cyc}\) is a group isomorphism \(\operatorname{Div}(X) \cong Z^1(X)\). Under this isomorphism, the subgroups of principal divisors on each side correspond to each other, whence it induces an isomorphism \(\operatorname{DivCl}(X) \cong \operatorname{Cl}(X)\).

Sketch of proof

Injectivity. If \(D\) is a Weil divisor or a Cartier divisor such that \(D\) and \(-D\) are effective, then \(D\) is trivial. It therefore suffices to show that the inverse image of the subset of effective Weil divisors under the homomorphism \(\operatorname{cyc}\) consists of effective Cartier divisors. So let \(D\) be a Cartier divisor on \(X\) such that \(\operatorname{cyc}(D)\) is effective. We can check that \(D\) is effective locally on \(X\), so we may assume that \(X = \operatorname{Spec}A\) for an integrally closed domain \(A\), and that \(D\) is principal, say given by \((X, f)\). By assumption \(f\in K(X)\) is contained in \(A_{\mathfrak p}\) for every \(\mathfrak p\in \operatorname{Spec}A\), and it follows from Lemma 3.8 Part (1) that \(f\in A\), as desired.

Surjectivity. We construct an inverse to the map \(\operatorname{cyc}\).

May 22,

If \(Z\subset X\) is an integral closed subscheme of \(X\) of codimension \(1\) with corresponding ideal sheaf \({\mathscr I}_Z\subseteq {\mathscr O}_X\), then for every \(x\in X\), \({\mathscr I}_{Z,x}\) is a principal ideal in \({\mathscr O}_{X,x}\) by Lemma 3.8 Part (2). Using Proposition 2.17, we find an affine open cover \((U_i)_i\) of \(X\) together with elements \(f_i\in K(X)\) such that \({\mathscr I}_{Z|U_i} = f_i{\mathscr O}_X\) (inside \({\mathscr K}_X\)), for each \(i\). We then map \([Z]\) to the Cartier divisor \((U_i, f_i)_i\) (well-defined since elements in a domain that generate the same principal ideal differ at most by a unit), and extend this construction to a map \(d\colon Z^1(X)\to \operatorname{Div}(X)\) by linearity. By construction we have \(\operatorname{cyc}\circ d = \operatorname{id}\), and this implies that \(\operatorname{cyc}\) is surjective (and hence bijective with inverse \(d\)).

We can phrase the definition of principal Weil divisor as saying that it is the image under \(\operatorname{cyc}\) of a principal Cartier divisor. It is therefore clear that we also obtain an isomorphism \(\operatorname{DivCl}(X)\cong \operatorname{Cl}(X)\).

We thus have identifications

\[ \operatorname{Pic}(X)\cong \operatorname{DivCl}(X)\cong \operatorname{Cl}(X). \]

Example 3.10
  1. For any UFD \(A\), \(\operatorname{Pic}(A) = 1\) as remarked above. In particular, all divisors on affine space \(\mathbb {A}^n_k\) over a field (or over any UFD) \(k\) are principal.

  2. Let \(k\) be a field. As shown on the problem sheets, \(\operatorname{Pic}(\mathbb {P}^1_k) \cong \mathbb {Z}\). We will see below that \(\operatorname{Pic}(\mathbb {P}^n_k)\cong \mathbb {Z}\) for every \(n\ge 1\).

The Picard group or equivalently the divisor class group of an integral scheme \(X\) contains interesting information about \(X\), but is often not easy to compute.

(3.5) The theorem of Riemann and Roch

No proofs were given in the lecture at this point for the following results.

Reference: [ H ] IV.1.

May 24,

Now let \(X\) be a Dedekind scheme which is a scheme of finite type over an algebraically closed field \(k\). In view of Proposition 3.9, we identify Cartier and Weil divisors. In addition we assume that \(X\) is projective, i.e., that there exist \(n\ge 1\) and a closed immersion \(X\hookrightarrow \mathbb {P}^n_k\).

For a (Weil) divisor \(D=\sum _Z n_Z[Z]\) we define the degree \(\deg (D)\) of \(D\) as \(\deg (D):=\sum _Z n_Z\). We obtain a group homomorphism \(Z^1(X) \to \mathbb {Z}\). Under our assumption that \(X\) is a closed subscheme of some projective space, one can show that this homomorphism factors through \(\operatorname{Cl}(X)\):

Theorem 3.11
Let \(f\in K(X)\). Then \(\deg (\mathop{\rm div}\nolimits (f)) = 0\).

This means that the degree homomorphism \(\operatorname{Div}(X)\to \mathbb {Z}\) factors through the divisor class group. In particular, we can speak of the degree of a line bundle, and we denote the degree of \({\mathscr L}\) by \(\deg ({\mathscr L})\).

To state the famous Theorem of Riemann–Roch, we introduce the following notation. For a divisor \(D\) we write \(\ell (D) = \dim _k \Gamma (X, {\mathscr O}_X(D))\).

Proposition 3.12
For each \(D\), \(\ell (D)\) is finite.

If \(\ell (D) \ge 0\), then \(\deg (D) \ge 0\). In fact, it was shown that whenever \(\ell (D)\ne 0\), then \(D\) is linearly equivalent to an effective divisor \(D'\), and then \(\deg (D) = \deg (D') \ge 0\).

Theorem 3.13 (Riemann-Roch)
For \(X\) as above, there exist \(g\in \mathbb {Z}_{\ge 0}\) and \(K \in \operatorname{Div}(X)\) such that for every divisor \(D\) on \(X\), we have
\[ \ell (D) - \ell (K-D) = \deg (D) + 1 -g. \]

Corollary 3.14
In the above situation, we have
  1. \(\ell (K) = g\),

  2. \(\deg (K) = 2g-2\),

  3. for every \(D\) with \(\deg (D) {\gt} 2g-2\), we have \(\ell (D) = \deg (D) + 1 -g\).


The corollary is easy to prove with the Theorem of Riemann-Roch at hand. In fact, for (1) use the theorem with \(D=0\) the trivial divisor, for (2) use \(D=K\), and for (3) use that under the assumption there \(\deg (K-D) {\lt} 0\), whence \(\ell (K-D)=0\), as remarked before.

The number \(g\) is called the genus of the curve \(X\). Part (3) of the corollary shows that it is uniquely determined by \(X\).

Remark 3.15
The linear equivalence class of the canonical divisor \(K\) is uniquely determined. In fact, assume that \(K\) and \(K'\) are divisors which both have the property of a canonical divisor as in the Riemann-Roch theorem. Using the theorem and the corollary, one computes that \(\ell (K-K') {\gt} 0\) and \(\ell (K'-K) {\gt} 0\). As was shown on the problem sheet, this implies that \({\mathscr O}_X(K-K')\) is trivial, or in other words that \(K\) and \(K'\) have the same divisor class.

For the projective line \(\mathbb {P}^1_k\), it is easy to prove the Theorem of Riemann-Roch by direct computations. It has genus \(0\). One can show that every \(X\) of genus \(0\) is isomorphic to \(\mathbb {P}^1_k\). (But if \(k\) is not assumed to be algebraically closed, then there may exist \(X\) as above of genus \(0\) which are not isomorphic to \(\mathbb {P}^1_k\).)

For \(X\) as above which is of the form \(V_+(f)\subset \mathbb {P}^2_k\), there is the following formula for the genus:

Proposition 3.16
Let \(X\) as above be of the form \(V_+(f)\subset \mathbb {P}^2_k\) for a homogeneous polynomial \(f\) of degree \(d\). Then the genus \(g\) of \(X\) is given by
\[ g = \frac{(d-1)(d-2)}{2}. \]

For example, elliptic curves (which are defined by a homogeneous polynomial of degree \(3\)) have genus \(1\).

(3.6) Line bundles on \(\mathbb {P}^n_k\)

References: [ GW1 ] , Ch. 8, Ch. 11, in particular Example 11.43, (8.5); [ H ] II.6, II.7.

We want to compute the Picard group of projective space over a field. To this end, we will use the following general proposition.

Proposition 3.17
Let \(X\) be a noetherian integral scheme such that all local rings \({\mathscr O}_{X, x}\) are unique factorization domains. Let \(U\subseteq X\) be an open subscheme, and let \(Z_1,\dots , Z_r\) be those irreducible components of \(X\setminus U\) that are of codimension \(1\) inside \(X\). We consider the \(Z_i\) as integral closed subschemes of \(X\). Then we have a short exact sequence
\[ 0\to \bigoplus _i \mathbb {Z}[Z_i]\to Z^1(X)\to Z^1(U)\to 0 \]
which induces an exact sequence
\[ \bigoplus _i \mathbb {Z}[Z_i]\to \operatorname{Cl}(X)\to \operatorname{Cl}(U)\to 0. \]


It is clear that we have the first short exact sequence when we think of integral closed subschemes in terms of their generic points. It is easy to check that the “restriction map” \(Z^1(X)\to Z^1(U)\) induces a homomorphism between the class groups, and this yields the second exact sequence.

May 31,

In terms of the identifications of the divisor class groups with the Picard groups of \(X\) and of \(U\), the map \(\operatorname{Pic}(X)\to \operatorname{Pic}(U)\) in the proposition is just the restriction of line bundles from \(X\) to the open subscheme \(U\).

Now let \(R\) be a ring and fix \(n\ge 1\). We cover \(\mathbb {P}^n_R\) by the standard charts \(U_i := D_+(X_i)\), as usual, and write \(U_{ij} := U_i\cap U_j\). For \(d\in \mathbb {Z}\), multiplication by the elements \((X_i/X_j)^d \in \Gamma (U_{ij}, {\mathscr O}_{\mathbb {P}^n_R})^\times \) defines isomorphisms \({\mathscr O}_{U_i|U_{ij}}\to {\mathscr O}_{U_j|U_{ij}}\) which give rise to a gluing datum of the \({\mathscr O}_{U_i}\)-modules \({\mathscr O}_{U_i}\). By gluing of sheaves, we obtain a line bundle \({\mathscr O}_{\mathbb {P}^n_R}(d)\). (Cf. Problems 23, 24, 25 in the case \(n=1\).) To shorten the notation, we sometimes just write \({\mathscr O}(d)\), when the space is clear from the context.

Lemma 3.18
We obtain a group homomorphism \(\mathbb {Z}\to \operatorname{Pic}(\mathbb {P}^n_R)\), \(d\mapsto {\mathscr O}(d)\).

Proposition 3.19
Writing \(R[X_0, \dots , X_n]_d\) for the submodule of homogeneous polynomials of degree \(d\) (with \(R[X_0, \dots , X_n]_d=0\) for \(d{\lt}0\)), we have natural isomorphisms
\[ \Gamma (\mathbb {P}^n_R, {\mathscr O}(d)) \cong R[X_0, \dots , X_n]_d \]
for all \(d\in \mathbb {Z}\).


We can make the gluing construction for \({\mathscr O}(d)\) explicit by identifying \(\Gamma (D_+(X_i), {\mathscr O}(d))\) with \(X_i^d R\left[\frac{X_0}{X_i}, \dots , \frac{X_n}{X_i}\right]\), and correspondingly

\[ \Gamma (D_+(X_iX_j), {\mathscr O}(d)) = X_i^d R\left[\frac{X_0}{X_i}, \dots , \frac{X_n}{X_i}, \frac{X_i}{X_j}\right] = X_j^d R\left[\frac{X_0}{X_j}, \dots , \frac{X_n}{X_j}, \frac{X_j}{X_i}\right] \]

(inside \(R[X_0, \dots , X_n, X_0^{-1}, \dots , X_n^{-1}]\)). This means that the restriction map is just the inclusion map. (If \(R\) is a domain, then this describes an embedding of \({\mathscr O}(d)\) into the constant sheaf \({\mathscr K}_{\mathbb {P}^n_R}\) with values the field of rational functions.)

With this description, since the relevant restriction maps are injective, we can identify \(\Gamma (\mathbb {P}^n_R, {\mathscr O}(d))\) with the intersection

\[ \bigcap _{i=0}^n X_i^d R\left[\frac{X_0}{X_i}, \dots , \frac{X_n}{X_i}\right]. \]

One checks that this intersection is \(R[X_0, \dots , X_n]_d\), as claimed.

Corollary 3.20
The above homomorphism \(\mathbb {Z}\to \operatorname{Pic}(\mathbb {P}^n_R)\), \(d\mapsto {\mathscr O}(d)\), is injective.

Now let \(R=k\) be a field (in fact, the same arguments apply to any noetherian unique factorization domain \(k\)). Then \(\mathbb {P}^n_k\) is a noetherian integral scheme all of whose local rings are unique factorization domains, so we can talk about Cartier divisors and about Weil divisors, and identify the two notions via the cycle map as in Proposition 3.9.

Corollary 3.21
Let \(k\) be a field and let \(n\ge 1\). Then \(\operatorname{Pic}(\mathbb {P}^n_k)\cong \mathbb {Z}\).


If we identify \(\operatorname{Cl}(\mathbb {P}^n_k) = \operatorname{Pic}(\mathbb {P}^n_k)\) and apply Proposition 3.17 to \(X=\mathbb {P}^n_k\) and \(U=D_+(X_0)\cong \mathbb {A}^n_k\), we obtain a surjection \(\mathbb {Z}\to \operatorname{Pic}(\mathbb {P}^n_k)\). The injectivity statement of the previous corollary implies that this surjection (which allos us to identify \(\operatorname{Pic}(\mathbb {P}^n_k)\) with some quotient of \(\mathbb {Z}\)) must be an isomorphism. (At this point we have not yet shown that the map \(d\mapsto {\mathscr O}(d)\) is surjective, and hence an isomorphism; this will follow from the discussion in the following section.)

Remark 3.22
One can show that every locally free \({\mathscr O}_{\mathbb {P}^1_k}\)-module is isomorphic to a direct sum of line bundles (Problem 27). Note though that this statement is not true for \(\mathbb {P}^n_k\), \(n{\gt}1\).

(3.7) Divisors on \(\mathbb {P}^n_k\)

Let \(k\) be a field (or more generally a noetherian unique factorization domain), and let \(n\ge 1\). Let us take a look at the line bundles \({\mathscr O}(d)\) from the point of view of Cartier divisors. Write

\[ \mathcal R = \{ f = \frac gh;\ g,h\in k[X_0, \dots , X_n]\ \text{non-zero homogeneous polynomials}\} . \]

For \(f = g/h\in \mathcal R\), we define \(\deg (f) = \deg (g)-\deg (h)\). We can identify \(K(X)^\times \) with the subgroup of \(\mathcal R\) of degree \(0\) elements.

Fix an element \(f\in \mathcal R\) and write \(d=\deg (f)\). Let \(D\) be the Cartier divisor \(\mathop{\rm div}\nolimits (f) := (D_+(X_i), f/X_i^d)_i\) (this is a new use of the symbol \(\mathop{\rm div}\nolimits \) since \(f\) is not an element of \(K(\mathbb {P}^n_k)\)). Describing the line bundle \({\mathscr O}_{\mathbb {P}^n_k}(D)\) in terms of a gluing datum, it follows that \({\mathscr O}_X(D)\cong {\mathscr O}(d)\). Thus the composition

\[ \mathcal R\to \operatorname{Div}(\mathbb {P}^n_k) \to \operatorname{Pic}(\mathbb {P}^n_k) \cong \mathbb {Z} \]

is the degree map on \(\mathcal R\). In particular, the isomorphism class of the line bundle \({\mathscr O}_{\mathbb {P}^n_k}(\mathop{\rm div}\nolimits (f))\) depends only on \(d\), not on the choice of \(f\).

Now let \(f\in k[X_0,\dots , X_n]\) be an irreducible homogeneous polynomial of degree \(d {\gt} 0\). Then \(V_+(f)\) is an integral closed subscheme of \(\mathbb {P}^n_k\) of codimension \(1\) (since the same is true after intersection with any of the open charts \(D_+(X_i)\) (unless the intersection is empty)). From the construction of the matching between Cartier and Weil divisors, one sees that the Cartier divisor \(\mathop{\rm div}\nolimits (f)\) defined above corresponds to the Weil divisor \([V_+(f)]\). As a particular example, for any fixed \(i\), the Weil divisor \([V_+(X_i)]\) of the line \(V_+(X_i)\) has associated line bundle \({\mathscr O}(1)\).

Since the identification of Cartier divisors with Weil divisors is a group isomorphism, one can extend this description to all divisors, by decomposing a general \(f\in \mathcal R\) as a product of irreducible homogeneous polynomials and of inverses of such polynomials.

Coming back to the case of an irreducible homogeneous polynomial \(f\) of degree \(d {\gt} 0\), the datum of the divisor \(\mathop{\rm div}\nolimits (f)\) corresponds to the choice of embedding of its associated line bundle \({\mathscr O}(d)\) into \({\mathscr K}_{\mathbb {P}^n_k}\). The image of this embedding contains the structure sheaf \({\mathscr O}_{\mathbb {P}^n_k}\), and going through the definitions shows that the global section \(1\in \Gamma (\mathbb {P}^n_k,{\mathscr O}_{\mathbb {P}^n_k})\) is mapped to \(f\in k[X_0,\dots , X_n]_d = \Gamma (\mathbb {P}^n_k, {\mathscr O}(d))\) (Proposition 3.19) under this embedding. Compare Problems 31, 33. In other words, the embedding \(\Gamma (\mathbb {P}^n_k, {\mathscr O}(d))\to K(X)\) of the global sections is given by

\[ k[X_0,\dots , X_n]_d\to K(X), \quad g\mapsto g/f. \]

June 7,

Corollary 3.23
Let \(k\) be a field, and let \(Z\subseteq \mathbb {P}^n_k\) be an integral closed subscheme of codimension \(1\). Then \(Z=V_+(f)\) for some homogeneous polynomial \(f\).


Consider the divisor \([Z]\) given by \(Z\). Viewed as a Cartier divisor, it corresponds to an embedding \({\mathscr O}_{\mathbb {P}^n_k}([Z])\to {\mathscr K}_{\mathbb {P}^n_k}\) whose image contains \({\mathscr O}_{\mathbb {P}^n_k}\) since the divisor \([Z]\) is effective. Let \(f\in k[X_0, \dots , X_n]_d\) be the image of \(1\in \Gamma (\mathbb {P}^n_k, {\mathscr O})\) in \(\Gamma (\mathbb {P}^n_k, {\mathscr O}([Z])) = k[X_0, \dots , X_n]_d\) under this embedding. Since the embedding \({\mathscr O}_{\mathbb {P}^n_k}([Z])\to {\mathscr K}_{\mathbb {P}^n_k}\) is entirely determined by this image, the above discussion shows that \([Z] = [V_+(f)]\) as divisors and hence that \(Z = V_+(f)\).

(3.8) Functorial description of \(\mathbb {P}^n\)

As we have seen in Section1.1, every scheme \(X\) defines a contravariant functor \(T\mapsto X(T):=\operatorname{Hom}_{{\rm (Sch)}}(T, X)\) from the category of schemes to the category of sets. This functor determines \(X\) up to unique isomorphism. In this section, we want to describe the functor attached in this way to projective space \(\mathbb {P}^n_R\) for \(R\) a ring.

Lemma 3.24
Let \(X\) be a scheme.
  1. Let \({\mathscr F}\) be an \({\mathscr O}_X\)-module. Giving an \({\mathscr O}_X\)-module homomorphism \(\alpha \colon {\mathscr o}_X^{n+1}\to {\mathscr F}\) is “the same” as giving global sections \(s_0,\dots , s_n\in \Gamma (X, {\mathscr F})\) (namely the images of the standard basis vectors of \(\Gamma (X, {\mathscr O}_X^{n+1}) = \Gamma (X, {\mathscr O}_X)^{n+1}\).

  2. Now let \({\mathscr L}\) be a line bundle on \(X\), and let \(\alpha \colon {\mathscr O}_X^{n+1}\to {\mathscr L}\) be an \({\mathscr O}_X\)-module homomorphism given by \(s_0, \dots , s_n\in \Gamma (X, {\mathscr L})\). Then \(\alpha \) is surjective if and only if for every \(x\in X\) there exists \(i\) such that \(s_i(x)\ne 0\) in the fiber \({\mathscr L}(x)\).

Proposition 3.25
Let \(R\) be a ring, and let \(S\) be an \(R\)-scheme. There are bijections, functorial in \(S\),
\begin{align*} \mathbb {P}^n_R(S) = \{ ({\mathscr L}, \alpha );\ & {\mathscr L}\ \text{a line bundle on}\ S,\\ & \alpha \colon {\mathscr O}_S^{n+1}\twoheadrightarrow {\mathscr L}\ \text{a surjective}\ {\mathscr O}_S\text{-module homom.} \} /\cong . \end{align*}
Here we consider pairs \(({\mathscr L}, \alpha )\), \(({\mathscr L}', \alpha ')\) as isomorphic, if there exists an \({\mathscr O}_S\)-module isomorphism \(\beta \colon {\mathscr L}\to {\mathscr L}'\) with \(\alpha = \alpha '\circ \beta \).

Saying that the bijections of the proposition are functorial means that given a morphism \(S'\to S\) of \(R\)-schemes the bijections for \(S\) and \(S'\) together with the natural map \(\mathbb {P}^n_R(S)\to \mathbb {P}^n_R(S')\) and the map \(({\mathscr L}, \alpha )\mapsto (g^*{\mathscr L}, g^*\alpha )\) give rise to a commutative diagram. (Note that for every line bundle \({\mathscr L}\) on \(S\) the pull-back \(g^*{\mathscr L}\) is a line bundle on \(S'\), and for surjective \(\alpha \) the pull-back \(g^*\alpha \) is again a surjective \({\mathscr O}_{S'}\)-homomorphism of the desired form.)


A homomorphism \(\alpha \colon {\mathscr O}_S^{n+1}\twoheadrightarrow {\mathscr L}\) corresponds to \(n+1\) global sections in \(\Gamma (S, {\mathscr L})\) (the “images of the standard basis vectors”). Thus \(X_0, \dots , X_n\in \Gamma (\mathbb {P}^n_R, {\mathscr O}(1))\) give rise to a homomorphism \({\mathscr O}_{\mathbb {P}^n_R}^{n+1}\to {\mathscr O}(1)\). This homomorphism is surjective. (In fact, looking back at the construction of \({\mathscr O}(1)\) by gluing and the way how we identified the global sections of \({\mathscr O}(1)\) with \(R[X_0, \dots , X_n]_1\), under the identification \({\mathscr O}(1)_{D_+(X_i)} \cong {\mathscr O}_{D_+(X_i)}\) the restriction of \(X_i\) to \(D_+(X_i)\) corresponds to \(1\in \Gamma (D_+(X_i), {\mathscr O}_{D_+(X_i)})\) and in particular is non-zero in every fiber.)

Given a morphism \(S\to \mathbb {P}^n_R\), we can pull this homomorphism back to \(S\) and obtain an element of the right hand side in the statement of the proposition.

Conversely, given a pair \(({\mathscr L}, \alpha )\) on \(S\), we can think of the corresponding morphism \(S\to \mathbb {P}^n_R\) in terms of homogeneous coordinates (i.e., for \(K\)-valued points for some field \(K\)), as follows: Denote by \(f_0, \dots , f_n\in \Gamma (S, {\mathscr L})\) the global sections corresponding to \(\alpha \). For a point \(x\in S\), the fiber \({\mathscr L}(x)\) is a one-dimensional \(\kappa (x)\)-vector space generated by the elements \(f_0(x), \dots , f_n(x)\) (i.e., at least one of them is \(\ne 0\) – this holds since \(\alpha \) is surjective). We choose an isomorphism \({\mathscr L}(x) \cong \kappa (x)\), and hence can view the \(f_i(x)\) as elements of \(\kappa (x)\). Then the morphism \(S\to \mathbb {P}^n_S\) maps \(x\) to \((f_0(x) : \cdots : f_n(x)) \in \mathbb {P}^n(\kappa (x))\). While the individual \(f_i(x)\), as elements of \(\kappa (x)\), depend on the choice of isomorphism \({\mathscr L}(x)\cong \kappa (x)\), the point \((f_0(x) : \cdots : f_n(x)) \in \mathbb {P}^n(\kappa (x))\) is independent of this choice.

To make this rigorous, consider a pair \(({\mathscr L}, \alpha )\) as above, and for \(i\in \{ 0,\dots , n\} \) define

\[ S_i = \{ s\in S;\ \alpha (e_i)\ \text{generates the fiber}\ {\mathscr L}(s)\} , \]

where \(e_i\in {\mathscr O}_S^{n+1}\) denotes the \(i\)-th standard basis vector. This defines an open cover of \(S\). By definition, composing \(\alpha \) with the injection \({\mathscr O}_S\to {\mathscr O}_S^{n+1}\) as the \(i\)-th summand induces a trivialization \({\mathscr O}_{S_i}\cong {\mathscr L}_{|S_i}\) of the restriction of \({\mathscr L}\). We obtain a morphism \(S_i\to D_+(X_i) = \operatorname{Spec}k[\frac{X_0}{X_i},\dots , \frac{X_n}{X_i}]\) by mapping \(\frac{X_j}{X_i}\) to the image of \(\alpha (e_j)\in \Gamma (S_i, {\mathscr L})\) under the isomorphism \(\Gamma (S_i, {\mathscr L}) \to \Gamma (S_i, {\mathscr O}_S)\). These morphisms can be glued, and one obtains the desired morphism \(S\to \mathbb {P}^n_R\).

To conclude the proof, one checks that the two constructions are inverse to each other.

Remark 3.26
If \(S = \operatorname{Spec}k\) for a field \(k\), then every locally free sheaf on \(S\) is free, and the proposition reads as
\[ \mathbb {P}^n(k) = \{ \alpha \colon k^{n+1}\to k\ \text{surjective} \} /\cong , \]
where now homomorphisms \(\alpha \), \(\alpha '\) are isomorphic if and only if they have the same kernel. Thus we can identify
\[ \mathbb {P}^n(k) = \{ U\subset k^{n+1}\ \text{sub-vector space of dimension}\ n \} . \]
This description is dual to the classical description of \(\mathbb {P}^n(k)\) as the set of lines in \(k^{n+1}\). Passing to the dual space, the projection \(k^{n+1}\to k^{n+1}/U\) induces an inclusion \((k^{n+1}/U)^\vee \to k^{n+1,\vee }\) of the dual vector spaces. Matching the standard basis of \(k^{n+1}\) with its dual basis, we can identify \(k^{n+1,\vee } = k^{n+1}\), and in this way we get back the description in terms of lines.