Situation 111.56.5 (Notation plane curve). Let $k$ be an algebraically closed field. Let $F(X_0, X_1, X_2) \in k[X_0, X_1, X_2]$ be an irreducible polynomial homogeneous of degree $d$. We let

$D = V(F) \subset \mathbf{P}^2$

be the projective plane curve given by the vanishing of $F$. Set $x = X_1/X_0$ and $y = X_2/X_0$ and $f(x, y) = X_0^{-d}F(X_0, X_1, X_2) = F(1, x, y)$. We denote $K$ the fraction field of the domain $k[x, y]/(f)$. We let $C$ be the abstract curve corresponding to $K$. Recall (from the lectures) that there is a surjective map $C \to D$ which is bijective over the nonsingular locus of $D$ and an isomorphism if $D$ is nonsingular. Set $f_ x = \partial f/\partial x$ and $f_ y = \partial f/\partial y$. Finally, we denote $\omega = \text{d}x/f_ y = - \text{d}y/f_ x$ the element of $\Omega _{K/k}$ discussed in the lectures. Denote $K_ C$ the divisor of zeros and poles of $\omega$.

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