Divisors, principal divisors and canonical divisors on a Riemann surface

Let $X$ be a Riemann surface (Riemann surfaces and holomorphic atlases). A divisor on $X$ is a function $D:X\to \mathbb{Z}$ whose support $\mathrm{supp}D:=\{p:D(p)\mathrm{\ne }0\}$ is locally finite: every point has a neighbourhood meeting the support in only finitely many points. Write $D={\sum }_{p\in X}D(p)[p]$. Divisors form the abelian group $\mathrm{Div}(X)$ under pointwise addition; $D$ is effective, written $D\ge 0$, if every coefficient is nonnegative, and $D\ge D^{\mathrm{\prime }}$ means $D-D^{\mathrm{\prime }}\ge 0$. If $X$ is compact, local finiteness and compactness imply that every divisor has finite support, and its degree is $\mathrm{deg}D:={\sum }_{p}D(p)\in \mathbb{Z}$.

For a meromorphic function $f$ near $p$, define ${\mathrm{ord}}_p(f)=+\mathrm{\infty }$ if its germ at $p$ is identically zero. Otherwise, in a coordinate $z$ with $z(p)=0$, write uniquely $f=z^{m}u$ with $m\in \mathbb{Z}$ and $u$ holomorphic and nonzero at $p$, and put ${\mathrm{ord}}_p(f):=m$. The value is independent of the coordinate, since a change of coordinate has the form $z^{\mathrm{\prime }}=zv(z)$ with $v(0)\mathrm{\ne }0$. For a nonzero meromorphic function on connected $X$, no germ is identically zero: the set of points where it vanishes on a neighbourhood is open and closed, by the local identity theorem after clearing any pole, so connectedness makes that set either empty or all of $X$ (Identity theorem for holomorphic functions). Its zeros and poles are isolated, so $(f):={\sum }_{p\in X}{\mathrm{ord}}_p(f)[p]$ is a divisor, called the principal divisor of $f$. At a zero of $f$ its order is the ramification index $e_p(f)$ of the map $f:X\to \widehat{\mathbb{C}}$ at $0$; at a pole it is $-e_p(f)$, using the target coordinate $1\mathrm{/}w$ at $\mathrm{\infty }$ (Holomorphic maps and meromorphic functions on Riemann surfaces, Local power-map normal form on Riemann surfaces, Ramification index, ramification order and branch value, Isolated singularities: removable, poles, and essential singularities). For nonzero meromorphic functions $f,g$, local orders add, so $(fg)=(f)+(g)$ and $(1\mathrm{/}f)=-(f)$.

When $X$ is compact, every nonconstant meromorphic function $f$ has degree-zero principal divisor. Indeed, $f:X\to \widehat{\mathbb{C}}$ is continuous and the target is Hausdorff because it is the Riemann sphere (Holomorphic maps and meromorphic functions on Riemann surfaces, Riemann surfaces and holomorphic atlases); if $K\subseteq \widehat{\mathbb{C}}$ is compact, then $K$ is closed, so $f^{-1}(K)$ is closed in compact $X$ and hence compact. Thus $f$ is proper. The degree theorem for proper holomorphic maps says that the sum of local degrees over each fibre is the same integer $d$ (In a Hausdorff space a point and a disjoint compact set, and two disjoint compact sets, have disjoint open neighbourhoods; hence every compact subset is closed, and in a compact Hausdorff space the compact subsets are exactly the closed ones, A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact, Degree of a proper holomorphic map of Riemann surfaces). Applied to the fibres over $0$ and $\mathrm{\infty }$, this gives ${\sum }_{f(p)=0}{\mathrm{ord}}_p(f)=d={\sum }_{f(p)=\mathrm{\infty }}-{\mathrm{ord}}_p(f)$, hence $\mathrm{deg}((f))=0$; a nonzero constant also has divisor $0$. Two divisors are linearly equivalent, written $D\sim D^{\mathrm{\prime }}$, when $D-D^{\mathrm{\prime }}$ is principal.

For a nonzero meromorphic differential $\omega$ on $X$, the order ${\mathrm{ord}}_p(\omega )$ is the order of its local meromorphic coefficient in a coordinate. This is coordinate-independent because the transition factor for a differential is the derivative of a coordinate change, a holomorphic unit. The canonical divisor of $\omega$ is $(\omega ):={\sum }_p{\mathrm{ord}}_p(\omega )[p]$. If ${\omega }_1,{\omega }_2$ are nonzero meromorphic differentials, the local quotients of their coefficients glue to a nonzero meromorphic function $f={\omega }_1\mathrm{/}{\omega }_2$, and $({\omega }_1)=(f)+({\omega }_2)$; therefore all canonical divisors are linearly equivalent (Meromorphic differentials, orders and residues).

For a divisor $D$, set $L(D):=\{0\}\cup \{f\in \mathcal{M}(X):f\mathrm{\ne }0\text{ and }(f)+D\ge 0\},$ where $\mathcal{M}(X)$ is the field of meromorphic functions on $X$. Equivalently, a nonzero $f$ lies in $L(D)$ exactly when ${\mathrm{ord}}_p(f)\ge -D(p)$ for every $p$. These local lower bounds are preserved under addition and scalar multiplication, so $L(D)$ is a $\mathbb{C}$-vector space. Put $\mathrm{\ell }(D):={\dim }_{\mathbb{C}}L(D)\in {\mathbb{N}}_0\cup \{\mathrm{\infty }\}$. If $D\sim D^{\mathrm{\prime }}$ and $g$ is a nonzero meromorphic function with $(g)=D^{\mathrm{\prime }}-D$, then multiplication by $1\mathrm{/}g$ gives an isomorphism $L(D)\to L(D^{\mathrm{\prime }})$, because for $f\in L(D)$, $(f\mathrm{/}g)+D^{\mathrm{\prime }}=(f)-(g)+D^{\mathrm{\prime }}=(f)+D\ge 0;$ its inverse is multiplication by $g$. The meromorphic functions on $X$ form a field under the usual local sum, product and reciprocal operations (Meromorphic functions on a plane domain). In particular, when $X$ is compact and $\mathrm{deg}D<0$, the space $L(D)$ is zero: a nonzero $f\in L(D)$ would make the effective divisor $(f)+D$ have degree $\mathrm{deg}((f))+\mathrm{deg}D=\mathrm{deg}D<0$, impossible.