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This section is devoted to the ``core'' of the presented algorithm,
the BrillNoether residue theorem. In an intuitive
formulation it states that ``the intersection divisors of
with all adjoint curves of a fixed degree form a complete linear system,
up to being shifted by the adjoint divisor''.
Let be a closed point of . We introduce the local (resp. semilocal) rings
, resp.
, where
denote the closed places of over . Note that
is a subring
of
. The conductor
is an ideal both in
and in
.
It defines a divisor
on
whose support is
, the set of all places of over the
singular point . Recall that such a place corresponds to a unique
branch of . By abuse of notation, we sometimes do not distinguish
between the place and the corresponding branch.
Definition 2.4
We call the divisor
the
adjunction
divisor (or the
divisor of double points)
of
. Its support is the set of all places
over singular points of
.
Remark 2.5
The adjunction divisor
is rational, that is,
conjugate branches
satisfy
.
We have
if
is a place over
an
irreducible plane curve singularity
, resp.

(2) 
if
corresponds to the local branch
of
(
denoting the
invariant and
the local intersection multiplicity at
).
Alternatively, one can use the
Dedekind formula to compute the
multiplicities
: let
be a
primitive parametrization of the branch
then
if the respective expressions are finite (notice that either
or
does not vanish identically).
Notation.
Let
be a homogeneous form of degree
such that does not divide .
Then we denote by the intersection divisor on
cut out by the (preimage under of the) plane
curve defined by .
Note that is an adjoint divisor iff the intersection multiplicity
of with every local branch of is at least .
Proposition 2.7 (BrillNoether residue theorem)
Let
be a
reduced absolutely irreducible plane projective curve
given by the homogeneous polynomial
. Moreover, let
be the
normalization and
a rational divisor on
.
Finally, let
be an adjoint form of degree
such that
. Then we can identify
under the isomorphism
induced by
.
This proposition is an immediate corollary
of
Theorem 2.8 (M. Noether)
Let
be
homogeneous forms such that
. Then
there exist homogeneous forms
such that
.
For a complete proof we refer to [42], pp. 215ff,
resp. [28], Prop. 4.1.
Remark 2.9
Haché [
18] has shown that a
form
as in the BrillNoether
residue theorem exists whenever

(3) 
where
denotes the effective part of the divisor
.
Next: Computational Aspects
Up: Effective Construction of Algebraic
Previous: Symbolic HamburgerNoether expressions
Christoph Lossen
20010321