Adjunction for smooth plane curves

(Set-up.) Since $C=V_{+}(F)$ has pure dimension one, $F\mathrm{\ne }0$: otherwise $C={\mathbf{P}}_k^2$. Thus the hypersurface sequence of [F1] is exact and $\mathcal{I}\cong {\mathcal{O}}_{{\mathbf{P}}^2}(-d)$; the smooth curve $C$ has pure codimension one in ${\mathbf{P}}_k^2$.

(The ideal is the twist.) By [F1] the multiplication map ${\mathcal{O}}_{{\mathbf{P}}^2}(-d)\to {\mathcal{O}}_{{\mathbf{P}}^2}$ has image $\mathcal{I}$, so it induces an isomorphism ${\mathcal{O}}_{{\mathbf{P}}^2}(-d)\stackrel{\sim }{\to }\mathcal{I}$ of ${\mathcal{O}}_{{\mathbf{P}}^2}$-modules.

(Regularity of the immersion.) Since $C$ is smooth of pure dimension one, the closed immersion $i$ is a regular immersion of pure codimension one in the smooth ambient scheme ${\mathbf{P}}_k^2$, so [F2] gives that the conormal sheaf $\mathcal{I}\mathrm{/}{\mathcal{I}}^2$ is a locally free ${\mathcal{O}}_C$-module of rank one and the conormal sequence is exact.

(The ambient dualizing bundle.) By [F4] the dualizing bundle of the plane is ${\omega }_{{\mathbf{P}}^2}\cong {\mathcal{O}}_{{\mathbf{P}}^2}(-3)$, so its restriction to $C$ is $i^{\ast }{\omega }_{{\mathbf{P}}^2}\cong i^{\ast }{\mathcal{O}}_{{\mathbf{P}}^2}(-3)={\mathcal{O}}_C(-3)$ by [F5].

(The conormal sheaf.) Tensoring the isomorphism of step 2.1 with the structure sheaf of $C$ along $i$ gives $i^{\ast }\mathcal{I}\cong i^{\ast }{\mathcal{O}}_{{\mathbf{P}}^2}(-d)={\mathcal{O}}_C(-d)$ by [F5]; for the invertible ideal $\mathcal{I}$ the conormal sheaf is $\mathcal{I}\mathrm{/}{\mathcal{I}}^2=\mathcal{I}{\otimes }_{{\mathcal{O}}_{{\mathbf{P}}^2}}{\mathcal{O}}_C=i^{\ast }\mathcal{I}$, because locally $\mathcal{I}=(f)$ and tensoring with ${\mathcal{O}}_C$ presents $\mathcal{I}\mathrm{/}f\mathcal{I}$; hence $\mathcal{I}\mathrm{/}{\mathcal{I}}^2\cong {\mathcal{O}}_C(-d)$ by step 2.2.

(The determinant of the normal bundle.) The normal bundle is ${\mathcal{N}}_{C\mathrm{/}{\mathbf{P}}^2}=(\mathcal{I}\mathrm{/}{\mathcal{I}}^2{)}^{\vee }$ by [F3], a rank-one invertible ${\mathcal{O}}_C$-module by step 3.1 and [F5], and for $c=1$ the determinant is $\det \mathcal{N}=\mathcal{N}$; dualizing $\mathcal{I}\mathrm{/}{\mathcal{I}}^2\cong {\mathcal{O}}_C(-d)$ and using that the dual of ${\mathcal{O}}_C(-d)$ is ${\mathcal{O}}_C(d)$ for an invertible sheaf gives $\det {\mathcal{N}}_{C\mathrm{/}{\mathbf{P}}^2}\cong {\mathcal{O}}_C(d)$.

(Adjunction.) Applying [F3] to the smooth curve $C$ of pure dimension one inside ${\mathbf{P}}_k^2$, of codimension $c=1$, gives ${\omega }_C\cong i^{\ast }{\omega }_{{\mathbf{P}}^2}{\otimes }_{{\mathcal{O}}_C}\det {\mathcal{N}}_{C\mathrm{/}{\mathbf{P}}^2}$; substituting step 2.3 and step 4.1 yields ${\omega }_C\cong {\mathcal{O}}_C(-3){\otimes }_{{\mathcal{O}}_C}{\mathcal{O}}_C(d)={\mathcal{O}}_C(d-3)$, which is the displayed bundle isomorphism.

(Canonical divisors.) A canonical divisor of $C$ is the divisor of a nonzero rational differential, that is, of a nonzero rational section of ${\omega }_C$; under the isomorphism of step 5.1 it is the divisor of a nonzero rational section of ${\mathcal{O}}_C(d-3)$, and any two such divisors differ by the divisor of the rational function relating the two sections, hence are linearly equivalent; this proves the consequence. Every use of choice above is inherited from the duality and projective-space suppliers cited in [F1]–[F5] through [F6].