Preliminaries, Notations

Throughout the following, let $\mathbf{F}$ be a finite field and $\overline{\mathbf{F}}$ an algebraic closure of $\mathbf{F}$. Moreover, let $C\subset \mathbb {P}^2(\mathbf{F})$ be an absolutely irreducible, reduced, projective plane curve given by a homogeneous form of degree $d$, $F\in \mathbf{F}[X,Y,Z]_d$.

A point $P\in C$ is called rational if its coordinates are in $\mathbf{F}$. More generally, by a closed point $[P]\in C$ we denote the formal sum of a point (defined over $\overline{\mathbf{F}}$) with its conjugates. If there is no risk of confusion, we sometimes write $P\in C$ to denote the closed point $[P]$. Note that closed points are invariant under the action of the Galois group $Gal(\overline{\mathbf{F}}/\mathbf{F})$.

We denote by $n:{\widetilde{C}}\to C$ the normalization and by

$$N:=i\circ n: \;{\widetilde{C}}\to \mathbb {P}^2(\mathbf{F})$$

the parametrization of $C$ (with $i:C\hookrightarrow \mathbb {P}^2(\mathbf{F})$ being the inclusion).

The points of ${\widetilde{C}}$ are called places of $C$. Again, a place is called rational if its coordinates are in $\mathbf{F}$, and by a closed place we denote the formal sum of a place (defined over $\overline{\mathbf{F}}$) with its conjugates. Note that each smooth (rational, resp. closed) point $P\in C$ corresponds to a unique (rational, resp. closed) place $n^{-1}(P)\in {\widetilde{C}}$. If $P$ is a singular point of $C$ then each local branch of $C$ at $P$ corresponds to a unique place of $C$. Hence, the set of places of $C$ can be identified with the set consisting of the non-singular points of $C$ and all tuples $\bigl(P;\!(C_i,P)\bigr)$, $P$ a singular point of $C$ and $(C_i,P)$ a local branch of $C$ at $P$.

A (rational) divisor $D$ on ${\widetilde{C}}$ is a finite, weighted, formal sum of (closed) places $Q$ of $C$, $D = \sum n_{Q} Q$ with integer coefficients $n_Q =: ord_Q (D)$. The divisor $D$ is called effective if there are no negative $n_Q$. Moreover, we introduce the degree of the divisor $D$, $\deg D:=\sum n_Q$ and the support of $D$, supp$\,D:=\{Q \mid n_Q\neq 0\}$.

To each element $g$ in the function field $\mathbf{F}({\widetilde{C}})$ of ${\widetilde{C}}$ one associates the principal divisor $(g):= \sum ord_Q(g)\cdot Q$. Note that $(g)$ has degree 0, by the residue theorem.

Finally, the linear system associated to a divisor $D$ is defined to be

$$\mathcal{L}(D)\,:=\, \bigl\{\, g\in \mathbf{F}({\widetilde{C}}) \, \mid \, (g)\geq -D\, \bigr\} \cup \bigl\{ 0 \bigr\}\,,$$

where the order relation is defined by

$$D\geq D' \;:\Longleftrightarrow \;\, ord_Q(D)\geq ord_Q(D') \,\text{ for all places $Q\in {\widetilde{C}}$}\,.$$