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## Adjoint curves and Brill-Noether residue theorem

This section is devoted to the core'' of the presented algorithm, the Brill-Noether 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 .

Definition 2.6   Let be a rational divisor on such that is not contained in the support of . We call an adjoint divisor of iff the pull-back divisor satisfies

Let , , such that . Then we call an adjoint form and the plane curve defined by an adjoint curve of .

Note that is an adjoint divisor iff the intersection multiplicity of with every local branch of is at least .

Proposition 2.7 (Brill-Noether 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 Brill-Noether residue theorem exists whenever

 (3)

where denotes the effective part of the divisor .

Next: Computational Aspects Up: Effective Construction of Algebraic Previous: Symbolic Hamburger-Noether expressions
Christoph Lossen
2001-03-21