Throughout the following, let be a finite field and an algebraic closure of . Moreover, let be an absolutely irreducible, reduced, projective plane curve given by a homogeneous form of degree , .
A point is called rational if its coordinates are in . More generally, by a closed point we denote the formal sum of a point (defined over ) with its conjugates. If there is no risk of confusion, we sometimes write to denote the closed point . Note that closed points are invariant under the action of the Galois group .
We denote by the normalization and by
The points of are called places of . Again, a place is called rational if its coordinates are in , and by a closed place we denote the formal sum of a place (defined over ) with its conjugates. Note that each smooth (rational, resp. closed) point corresponds to a unique (rational, resp. closed) place . If is a singular point of then each local branch of at corresponds to a unique place of . Hence, the set of places of can be identified with the set consisting of the non-singular points of and all tuples , a singular point of and a local branch of at .
A (rational) divisor on is a finite, weighted, formal sum of (closed) places of , with integer coefficients . The divisor is called effective if there are no negative . Moreover, we introduce the degree of the divisor , and the support of , supp.
To each element in the function field of one associates the principal divisor . Note that has degree 0, by the residue theorem.
Finally, the linear system associated to a divisor is defined to be