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