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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 $ C$ with all adjoint curves of a fixed degree $ m$ form a complete linear system, up to being shifted by the adjoint divisor''.

Let $ P$ be a closed point of $ C$. We introduce the local (resp. semilocal) rings $ \mathcal{O}:=\mathcal{O}_{C,P}$, resp. $ \overline{\mathcal{O}}:=(n_{\ast}\mathcal{O}_{{\widetilde{C}}})_P\cong
\prod_{i=1}^r \mathcal{O}_{{\widetilde{C}},Q_i}$, where $ Q_1,\dots,Q_r$ denote the closed places of $ C$ over $ P$. Note that $ \mathcal{O}$ is a subring of $ \overline{\mathcal{O}}$. The conductor

$\displaystyle \mathcal{C}_P :=\, \mathcal{C}_{\,\overline{\mathcal{O}}/\mathcal...{O}} \,\mid\, \phi\cdot \overline{\mathcal{O}} \subset \mathcal{O}

is an ideal both in $ \mathcal{O}$ and in $ \overline{\mathcal{O}}$. It defines a divisor $ \mathcal{A}_P$ on $ {\widetilde{C}}$ whose support is $ \{Q_1,\dots,Q_r\}$, the set of all places of $ C$ over the singular point $ P$. Recall that such a place corresponds to a unique branch of $ (C,P)$. By abuse of notation, we sometimes do not distinguish between the place and the corresponding branch.

Definition 2.4   We call the divisor $ \mathcal{A}:=\sum_P \mathcal{A}_P$ the adjunction divisor (or the divisor of double points) of $ C$. Its support is the set of all places over singular points of $ C$.

Remark 2.5   The adjunction divisor $ \mathcal{A}=\sum_Q d_Q Q$ is rational, that is, conjugate branches $ Q,Q'$ satisfy $ d_Q=d_{Q'}$. We have $ d_Q=2\cdot \delta(C,P)$ if $ Q$ is a place over an irreducible plane curve singularity $ (C,P)$, resp.

$\displaystyle d_Q\,=\,2\cdot \delta(C_i,P)+\sum_{j\neq i} i_P(C_i,C_j)$ (2)

if $ Q$ corresponds to the local branch $ (C_i,P)$ of $ (C,P)$ ($ \delta$ denoting the $ \delta$-invariant and $ i_P$ the local intersection multiplicity at $ P$). Alternatively, one can use the Dedekind formula to compute the multiplicities $ d_Q$: let $ \varphi_Q:\mathbf{F}[[x,y]]\to \overline{\mathbf{F}}[[t]]$ be a primitive parametrization of the branch $ Q$ then

$\displaystyle d_Q\,=\:ord_t \left(\frac{f_y\bigl(\varphi_Q(x),\varphi_Q(y)\bigr...
\frac{d}{dt} \!\; \varphi_Q(y)}\right) $

if the respective expressions are finite (notice that either $ \frac{d}{dt}\!\;\varphi_Q(x)$ or $ \frac{d}{dt} \!\;\varphi_Q(y)$ does not vanish identically).

Notation. Let $ H\in \mathbf{F}[X,Y,Z]_m$ be a homogeneous form of degree $ m$ such that $ F$ does not divide $ H$. Then we denote by $ N^{\ast}H$ the intersection divisor on $ {\widetilde{C}}$ cut out by the (preimage under $ N$ of the) plane curve defined by $ H$.

Definition 2.6   Let $ D$ be a rational divisor on $ \mathbb {P}^2(\mathbf{F})$ such that $ C$ is not contained in the support of $ D$. We call $ D$ an adjoint divisor of $ C$ iff the pull-back divisor satisfies $ N^{\ast}D \geq \mathcal{A}.$

Let $ H\in \mathbf{F}[X,Y,Z]_m$, $ F\!\not\vert\, H$, such that $ N^{\ast}H\geq \mathcal{A}$. Then we call $ H$ an adjoint form and the plane curve defined by $ H$ an adjoint curve of $ C$.

Note that $ D$ is an adjoint divisor iff the intersection multiplicity of $ D$ with every local branch $ Q$ of $ C$ is at least $ d_Q$.

Proposition 2.7 (Brill-Noether residue theorem)   Let $ C\stackrel{i}{\hookrightarrow}\mathbb {P}^2(\mathbf{F})$ be a reduced absolutely irreducible plane projective curve given by the homogeneous polynomial $ F\in \mathbf{F}[X,Y,Z]_d$. Moreover, let $ n:{\widetilde{C}}\to C$ be the normalization and $ D$ a rational divisor on $ {\widetilde{C}}$.

Finally, let $ H_0\in \mathbf{F}[X,Y,Z]_m$ be an adjoint form of degree $ m>0$ such that $ N^{\ast}H_0\geq
{\mathcal{A}}+D$. Then we can identify

$\displaystyle \mathcal{L}(D)\,\equiv\, \left\{ \left[\frac{H}{H_0}\right] \in \...
N^{\ast}H\geq N^{\ast}H_0\!-D
\end{array}\right\} \cup \bigl\{0 \bigr\} $

under the isomorphism $ \mathbf{F}({\widetilde{C}})\cong \mathbf{F}(C)$ induced by $ N$.

This proposition is an immediate corollary of

Theorem 2.8 (M. Noether)   Let $ G,H\in \mathbf{F}[X,Y,Z]$ be homogeneous forms such that $ N^{\ast}H\geq
{\mathcal{A}}+ N^{\ast}G$. Then there exist homogeneous forms $ A,B\in \mathbf{F}[X,Y,Z]$ such that $ H=AF\!\!\:+\!\!\:BG$.

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 $ H_0\in \mathbf{F}[X,Y,Z]_m$ as in the Brill-Noether residue theorem exists whenever

$\displaystyle m \,>\, \max\left\{\,d\!\!\:- \!\!\:1\,,\:\, \frac{d\!\!\:-\!\!\:3}{2}+ \frac{\deg\,(\mathcal{A}+\, D_{+})}{d}\right\}\,,$ (3)

where $ D_{+}$ denotes the effective part of the divisor $ D=D_{+}\!-D_{-}$.

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