2 Line bundles and divisors

General references:  [ GW ] 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


(2.1) 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 2.1
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 $i$, $j$.

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.

Definition 2.2
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 o$X$.

(2.2) 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 2.3
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 $\operatorname{DivCl}(X) \cong \operatorname{Pic}(X)$.

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 2.4
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


April 24,

Now let $X$ be a noetherian integral scheme such that all local rings ${\mathscr O}_{X,x}$ are factorial.

(2.3) 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 2.5
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 $Z$. Weil divisors of this form are called principal Weil divisors. Two Weil divisors are called linearly equivalent, if their difference is a principal divisor.

(2.4) 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 $).

Proposition 2.6
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 $\operatorname{DivCl}(X) \cong \operatorname{Cl}(X) \cong \operatorname{Pic}(X)$.

(2.5) The theorem of Riemann and Roch

No proofs were given in the lecture for the following results.

Reference: [ H ] IV.1.

Now let $X$ a Dedekind scheme which is a scheme of finite type over an algebraically closed field $k$. In addition we assume 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 2.7
Let $f\in K(X)$. Then $\deg (\mathop{\rm div}\nolimits (f)) = 0$.

We can now state (a simplified version of) the Theorem of Riemann–Roch. For a divisor $D$ we write $\ell (D) = \dim _k \Gamma (X, {\mathscr O}_X(D))$.

Proposition 2.8
For each $D$, $\ell (D)$ is finite. If $\ell (D) \ge 0$, then $\deg (D) \ge 0$.

Theorem 2.9 (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 2.10
In the above situation, we have
  1. $\ell (K) = g$,

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

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

The number $g$ is called the genus of the curve $X$. 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 2.11
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
\[ g = \frac{(d-1)(d-2)}{2}. \]