Adjunction on a smooth plane quartic: the canonical bundle is the hyperplane bundle

Given: AC, a field $k$ and a smooth plane quartic $C=V_{+}(F)\subseteq {\mathbb{P}}_k^2$ of degree $d=4$.

For a smooth plane curve $C=V_{+}(F)\subseteq {\mathbb{P}}_k^2$ of degree $d$, ${\omega }_C\cong {\mathcal{O}}_C(d-3)$ and $g(C)=(d-1)(d-2)\mathrm{/}2$; the degree of the hypersurface is $\mathrm{deg}F=d$. (Adjunction for smooth plane curves, The genus of a smooth plane curve in terms of its degree, degree projective hypersurface)

For a smooth proper geometrically integral curve of genus $g$, ${\mathrm{deg}}_k{\omega }_C=2g-2$ and $h^0(C,{\omega }_C)=g$, and, when the complete canonical system $\mathrm{\mid }{K}_C\mathrm{\mid }$ is base-point-free and its section space is nonzero, it defines the canonical morphism (The canonical divisor has degree 2g - 2, The canonical bundle has exactly g independent sections, Canonical bundle and canonical divisors, Complete linear system, A base-point-free linear system defines a morphism to projective space).

On ${\mathbb{P}}_k^2$ one has $H^1({\mathbb{P}}^2,\mathcal{O}(m))=0$ for every $m$, $H^0({\mathbb{P}}^2,\mathcal{O}(-3))=0$, and $H^0({\mathbb{P}}^2,\mathcal{O}(1))$ is the space of linear forms; the twisting sheaves are the ones attached to the standard graded presentation. (Cohomology of O(d) on projective space, Twisting sheaf on Proj)

If $i:C\hookrightarrow {\mathbb{P}}_k^2$ is the plane quartic, the published hypersurface sequence gives $0\to {\mathcal{O}}_{{\mathbb{P}}^2}(-4)\stackrel{\cdot F}{\to }{\mathcal{O}}_{{\mathbb{P}}^2}\to i_{\ast }{\mathcal{O}}_C\to 0$: the nonzero dehomogenizations of $F$ are nonzerodivisors in the polynomial domain rings of the standard charts. The standard graded polynomial ring generated in degree one makes ${\mathcal{O}}_{{\mathbb{P}}^2}(1)$ invertible, so tensoring preserves exactness. On each standard chart the quotient by the local equation $F$ with the restricted twist identifies the last term with $i_{\ast }{\mathcal{O}}_C(1)$, yielding $0\to {\mathcal{O}}_{{\mathbb{P}}^2}(-3)\stackrel{\cdot F}{\to }{\mathcal{O}}_{{\mathbb{P}}^2}(1)\to i_{\ast }{\mathcal{O}}_C(1)\to 0$. (Hypersurface cohomology sequence, Invertible twists for degree-one generated rings, Relative projective space from standard charts, Closed immersions are affine quotients and survive base change, Direct image of a sheaf along a continuous map, Twisting sheaf on Proj)

The short exact sequence in [F4] gives a long exact sequence in sheaf cohomology, and closed-immersion pushforward identifies $H^q({\mathbb{P}}^2,i_{\ast }{\mathcal{O}}_C(1))$ with $H^q(C,{\mathcal{O}}_C(1))$. In particular its low-degree segment is $0\to H^0({\mathbb{P}}^2,\mathcal{O}(-3))\to H^0({\mathbb{P}}^2,\mathcal{O}(1))\to H^0(C,{\mathcal{O}}_C(1))\to H^1({\mathbb{P}}^2,\mathcal{O}(-3))$. (Long exact sequence of sheaf cohomology, Closed immersion preserves cohomology and coherent pushforward)

For a smooth proper geometrically integral curve of genus $g\ge 2$, the canonical map is a closed immersion exactly when $g\ge 3$ and the curve is geometrically nonhyperelliptic, meaning that its algebraic-closure base change has no degree-two map to ${\mathbb{P}}^1$. (The canonical map: base-point-freeness and the hyperelliptic exception, Hyperelliptic curves and hyperelliptic maps)

For a divisor $D$ on a smooth proper curve, the complete linear system $\mathrm{\mid }D\mathrm{\mid }$ is the set of effective divisors linearly equivalent to $D$, and its dimension is $\mathrm{\ell }(D)-1$. (Complete linear system, Degree divisor proper curve)