# 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**

*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

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

**Definition 2.2**

*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:

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**

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**

*support*of a Cartier divisor $D$ is

## Weil divisors

April 24,

2019

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**

*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

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:

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**

**(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**

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**

**Theorem 2.9**(Riemann-Roch)

**Corollary 2.10**

$\ell (K) = g$,

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

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**