[proofplan] The proof combines two computations of the degree of the canonical divisor $K_C$ of $C$. From the adjunction-type formula for plane curves (a structural fact about the canonical sheaf of a smooth hypersurface in projective space), the canonical divisor of a smooth plane curve of degree $d$ is linearly equivalent to $(d - 3) H|_C$, where $H$ is a hyperplane (i.e., a line in $\mathbb{P}^2_k$). The hyperplane section $H|_C$ has degree $d$ on $C$ by the [Degree of Hyperplane Sections](/theorems/2178) — this is the definition of the degree of $C$ as a projective curve. Hence $\deg K_C = (d - 3) \cdot d = d^2 - 3d$. On the other hand, [Degree of the Canonical Divisor](/theorems/2186) gives $\deg K_C = 2g - 2$. Equating the two expressions and solving for $g$ yields the binomial formula. [/proofplan] [step:Identify the canonical divisor of $C$ via the adjunction formula for plane curves] We use the following structural fact about the canonical bundle of a smooth plane curve, the \emph{adjunction formula} for hypersurfaces in projective space: [claim:For a smooth projective irreducible plane curve $C \subset \mathbb{P}^2_k$ of degree $d$, the canonical divisor $K_C$ is linearly equivalent to $(d-3) H|_C$, where $H \subset \mathbb{P}^2_k$ is any line not contained in $C$.] [proof] This is the *Adjunction Formula* specialised to smooth plane curves. The general statement is: for a smooth hypersurface $X = V(F) \subset \mathbb{P}^n_k$ of degree $d$, the canonical bundle of $X$ is \begin{align*} \omega_X \cong \omega_{\mathbb{P}^n_k}|_X \otimes \mathcal{O}_{\mathbb{P}^n_k}(d)|_X = \mathcal{O}_{\mathbb{P}^n_k}(-n-1)|_X \otimes \mathcal{O}_{\mathbb{P}^n_k}(d)|_X = \mathcal{O}_{\mathbb{P}^n_k}(d - n - 1)|_X. \end{align*} For $n = 2$ (the case of a plane curve $C \subset \mathbb{P}^2_k$), this specialises to $\omega_C \cong \mathcal{O}_{\mathbb{P}^2_k}(d - 3)|_C$. Translating the line bundle isomorphism into a divisor statement: the canonical divisor $K_C$ is linearly equivalent to the divisor class of a hyperplane section of $\mathbb{P}^2_k$ scaled by $d - 3$, i.e., \begin{align*} K_C \sim (d - 3) H|_C \end{align*} where $H \subset \mathbb{P}^2_k$ is any line (i.e., hyperplane in $\mathbb{P}^2_k$) and $H|_C := \operatorname{div}(L|_C)$ is the divisor cut out on $C$ by a linear form $L$ defining $H$, provided $C \not\subset H$ (which is automatic since $C$ has dimension $1$ and $H$ has dimension $1$ in the same $\mathbb{P}^2_k$, so $C \subset H$ would force $C = H$, contradicting $\deg C = d$ vs $\deg H = 1$ unless $d = 1$). The verification of the adjunction formula for plane curves uses the conormal exact sequence. Writing $C = V(F)$ for $F \in k[X_0, X_1, X_2]_d$ a homogeneous polynomial of degree $d$ defining the smooth curve $C$, the conormal sheaf of $C$ in $\mathbb{P}^2_k$ is the line bundle $\mathcal{O}_{\mathbb{P}^2_k}(-d)|_C$ (generated locally by $F$ as a degree-$d$ form vanishing on $C$, smoothness of $C$ ensuring that $F$ generates the ideal sheaf as a line bundle). The conormal exact sequence (short exact, not split in general) \begin{align*} 0 \to \mathcal{O}_C(-d) \to \Omega_{\mathbb{P}^2_k}|_C \to \Omega_C \to 0 \end{align*} is a short exact sequence of locally free sheaves on $C$ (both ends are locally free of ranks $1$ and $1$, middle of rank $2$). Taking determinants of locally free sheaves in a short exact sequence $0 \to \mathcal{F} \to \mathcal{G} \to \mathcal{H} \to 0$ gives $\det \mathcal{G} = \det \mathcal{F} \otimes \det \mathcal{H}$: \begin{align*} \det \Omega_{\mathbb{P}^2_k}|_C = \mathcal{O}_C(-d) \otimes \det \Omega_C = \mathcal{O}_C(-d) \otimes \omega_C. \end{align*} Using $\det \Omega_{\mathbb{P}^2_k} = \omega_{\mathbb{P}^2_k} = \mathcal{O}_{\mathbb{P}^2_k}(-3)$ (computed from the Euler exact sequence for projective space), we get \begin{align*} \mathcal{O}_C(-3) = \mathcal{O}_C(-d) \otimes \omega_C, \quad\text{i.e.,}\quad \omega_C = \mathcal{O}_C(d - 3). \end{align*} Translating to divisors: $K_C \sim (d - 3) H|_C$ as claimed. [/proof] [/claim] We accept the claim as a structural input from the theory of canonical sheaves of smooth hypersurfaces (specifically, the adjunction formula). [/step] [step:Compute $\deg(H|_C) = d$ using the definition of the degree of a projective curve] Let $H \subset \mathbb{P}^2_k$ be any line not contained in $C$ (such a line exists because $C$ is a single curve of degree $d$ in $\mathbb{P}^2_k$ and there is a two-parameter family of lines, only finitely many of which could be components of $C$; concretely, since $C$ is irreducible of degree $d \geq 1$ and a line has degree $1$, $L \subset C$ is impossible unless $d = 1$ and $H = C$ — avoided by choosing $H \neq C$). Let $L \in k[X_0, X_1, X_2]_1$ be the linear form defining $H = V(L)$, and let $\operatorname{div}(L|_C)$ be the hyperplane-section divisor on $C$, i.e.\ the divisor of the restriction of $L$ regarded as a section of $\mathcal{O}_{\mathbb{P}^2_k}(1)|_C$. By [Degree of Hyperplane Sections](/theorems/2178), the degree of the projective curve $C$ equals the common value $\deg \operatorname{div}(L|_C)$ for any linear form $L$ with $C \not\subset V(L)$: \begin{align*} \deg(H|_C) = \deg \operatorname{div}(L|_C) = \deg(C) = d. \end{align*} This is the definition of the degree of a projective curve via hyperplane sections (theorem 2178), not Bézout's theorem for two curves; the equality is immediate from the definition and requires no multiplicity input beyond that embedded in $\deg \operatorname{div}$. Therefore the divisor $(d - 3) H|_C$ has degree \begin{align*} \deg\bigl((d - 3) H|_C\bigr) = (d - 3) \cdot d = d^2 - 3d. \end{align*} [guided] The degree of $H|_C$ on $C$ is the number of intersection points of the line $H$ with the curve $C$, counted with multiplicity. For a generic line, this is $d$ distinct points (a "generic" Bézout count); for special lines (tangent or higher-order tangent to $C$), some intersections coalesce but the total count with multiplicity is still $d$. \textbf{Why is this the definition of $\deg(C)$?} The degree of a projective curve $C \subset \mathbb{P}^N_k$ is, by [Degree of Hyperplane Sections](/theorems/2178), the common value $\deg \operatorname{div}(L|_C)$ for any linear form $L$ with $C \not\subset V(L)$. This common value is well-defined (independent of $L$) by Theorem 2178, which we proved earlier. For a plane curve, this is the number of intersections with a generic line. \textbf{Linear equivalence of canonical divisors.} The expression $K_C \sim (d-3) H|_C$ is a linear equivalence — the two divisors differ by the principal divisor of some rational function. Linear equivalence preserves degree (because principal divisors have degree zero by [Principal Divisors Have Degree Zero](/theorems/2177)), so $\deg K_C = \deg((d-3) H|_C) = (d-3) d$ unambiguously. [/guided] [/step] [step:Combine with the formula $\deg K_C = 2g - 2$ and solve for $g$] By [Degree of the Canonical Divisor](/theorems/2186), for any smooth projective irreducible curve $C$ of genus $g$ over an algebraically closed field, \begin{align*} \deg K_C = 2g - 2. \end{align*} Combined with the value $\deg K_C = d^2 - 3d$ from Step 2: \begin{align*} 2g - 2 = d^2 - 3d. \end{align*} Solving for $g$: \begin{align*} 2g = d^2 - 3d + 2 = (d - 1)(d - 2), \end{align*} hence \begin{align*} g = \frac{(d - 1)(d - 2)}{2} = \binom{d - 1}{2}. \end{align*} This completes the proof. [guided] The formula $g = \binom{d-1}{2}$ is one of the most striking results in algebraic geometry: a single binomial coefficient computes the genus of \emph{every} smooth plane curve in terms of its degree alone. Some checks: \begin{itemize} \item $d = 1$: $g = \binom{0}{2} = 0$. A line $C \cong \mathbb{P}^1_k$ has genus $0$. \checkmark \item $d = 2$: $g = \binom{1}{2} = 0$. A smooth conic is isomorphic to $\mathbb{P}^1_k$ (rational), genus $0$. \checkmark \item $d = 3$: $g = \binom{2}{2} = 1$. Smooth cubics in $\mathbb{P}^2_k$ are elliptic curves of genus $1$. \checkmark \item $d = 4$: $g = \binom{3}{2} = 3$. Smooth quartic plane curves have genus $3$. \item $d = 5$: $g = \binom{4}{2} = 6$. Smooth quintics have genus $6$. \end{itemize} \textbf{Why does the genus jump?} Each increment in $d$ adds $\binom{d-1}{2} - \binom{d-2}{2} = d - 2$ to the genus. So increasing degree by $1$ adds $d - 2$ to the genus, which grows linearly. The total genus grows quadratically: $g \sim d^2/2$ for large $d$. \textbf{Why does this argument work only for \emph{smooth} plane curves?} The adjunction formula $K_C = (d - 3) H|_C$ uses smoothness of $C$ in its derivation (the conormal sheaf is a line bundle iff $C$ is a smooth divisor). For singular plane curves, the formula breaks: the genus is generally lower than $\binom{d-1}{2}$, with the discrepancy controlled by the local contributions of singularities (the so-called $\delta$-invariants). The smooth case is the maximal-genus case for plane curves of given degree. \textbf{Embedding obstructions.} Not every smooth projective curve embeds in $\mathbb{P}^2_k$. Curves of genera $g = 0, 1, 3, 6, 10, \ldots$ (the values $\binom{d-1}{2}$ for $d = 1, 2, 3, 4, 5, \ldots$) admit smooth plane models, but generic curves of other genera (e.g., $g = 2, 4, 5, 7, 8, 9$) do not. Determining which curves embed in $\mathbb{P}^2_k$ — and how many ways — is a deep classical question. [/guided] [/step]