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.

With addition given by

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

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

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 $.

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.

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 $).

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 $\operatorname{Cl}(X)$:

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

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: