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

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

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.

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

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

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

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

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

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

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