[proofplan] The strategy is a direct application of the [Embedding Criterion](/theorems/2195) via the [Large Degree Implies Condition $(*)$](/theorems/2196) shortcut. We first compute $\deg(3K_C) = 6g - 6$ using $\deg K_C = 2g - 2$ from the [Degree of the Canonical Divisor](/theorems/2186). For $g \geq 2$ this satisfies the strict inequality $6g - 6 > 2g$, which is the hypothesis of [Large Degree Implies Condition $(*)$](/theorems/2196) — so $3K_C$ satisfies condition $(*)$. The [Embedding Criterion](/theorems/2195) then gives that $\phi_{3K_C}$ is an embedding. We separately compute that the target projective space has dimension $5g - 6$ by applying [Riemann–Roch for Large Degree Divisors](/theorems/2188) to $3K_C$. [/proofplan] [step:Compute $\deg(3K_C) = 6g - 6$ from the canonical divisor degree formula] By the [Degree of the Canonical Divisor](/theorems/2186), the canonical divisor $K_C$ of a smooth projective curve of genus $g$ has degree \begin{align*} \deg K_C = 2g - 2. \end{align*} The degree map on the divisor group $\operatorname{Div}(C)$ is a group homomorphism into $\mathbb{Z}$ (it sends $D = \sum n_p \cdot p$ to $\sum n_p$), so it is in particular $\mathbb{Z}$-linear: \begin{align*} \deg(3 K_C) = 3 \cdot \deg(K_C) = 3 (2g - 2) = 6g - 6. \end{align*} [/step] [step:Verify the strict inequality $\deg(3K_C) > 2g$ using $g \geq 2$] We must check the hypothesis $\deg D > 2g$ of the theorem [Large Degree Implies Condition $(*)$](/theorems/2196), with $D := 3K_C$. Substituting $\deg(3K_C) = 6g - 6$: \begin{align*} \deg(3 K_C) - 2g = (6g - 6) - 2g = 4g - 6. \end{align*} For $g \geq 2$, we have $4g - 6 \geq 4 \cdot 2 - 6 = 2 > 0$. Hence \begin{align*} \deg(3 K_C) = 6g - 6 > 2g, \end{align*} strictly. The hypothesis of [Large Degree Implies Condition $(*)$](/theorems/2196) is satisfied. [guided] The threshold $\deg D > 2g$ is the natural cut-off above which Riemann–Roch becomes "trivial" (the correction term $\ell(K_C - D)$ vanishes), and above which a divisor automatically separates points and tangents. Here we are checking that taking three copies of the canonical divisor pushes us past this threshold for every curve with $g \geq 2$. \textbf{Why $g \geq 2$ matters.} The inequality $4g - 6 > 0$ is equivalent to $g > 3/2$, i.e.\ $g \geq 2$ for integer-valued $g$. So $g = 2$ is exactly the borderline — and one verifies $4(2) - 6 = 2 > 0$, so $g = 2$ is included. For $g = 0$ we get $4(0) - 6 = -6 < 0$, and for $g = 1$ we get $4(1) - 6 = -2 < 0$, so the inequality fails. (Of course, $\deg K_C = 2g - 2$ is non-positive for $g \leq 1$, and the geometry is genuinely different there: $g = 0$ has $C \cong \mathbb{P}^1_k$ and $g = 1$ has $C$ an elliptic curve, neither of which admit a canonical embedding in the sense considered here.) \textbf{Why $3K_C$ rather than $K_C$ or $2K_C$?} The [Canonical Embedding Theorem](/theorems/2197) says $\phi_{K_C}$ is an embedding iff $C$ is non-hyperelliptic. The bicanonical $\phi_{2K_C}$ also fails for some low-genus hyperelliptic curves. The point of considering $3K_C$ is precisely that the strict inequality $\deg(3K_C) = 6g - 6 > 2g$ holds \emph{universally} for $g \geq 2$, with no exceptions for hyperelliptic curves. So the tricanonical map is a uniform construction giving an embedding for \emph{every} smooth projective curve of genus at least $2$. This is exactly the role $3K_C$ plays in the proofs of the moduli space of curves and Mumford's GIT construction. [/guided] [/step] [step:Apply the large-degree shortcut to conclude $3K_C$ satisfies condition $(*)$] By [Large Degree Implies Condition $(*)$](/theorems/2196), if $D$ is any divisor on $C$ with $\deg D > 2g$, then $D$ satisfies condition $(*)$ — that is, for every pair of points $p, q \in C$ (allowing $p = q$), \begin{align*} \ell(D - p - q) = \ell(D) - 2. \end{align*} The hypotheses of this theorem are: \begin{itemize} \item $C$ is a smooth projective curve over an algebraically closed field $k$ — given. \item $D$ is a divisor on $C$ with $\deg D > 2g$ — verified in Step 2 with $D = 3K_C$. \end{itemize} Both hypotheses hold. We conclude that $3K_C$ satisfies condition $(*)$. [/step] [step:Apply the embedding criterion to conclude $\phi_{3K_C}$ is an embedding] By the [Embedding Criterion](/theorems/2195), the linear-system morphism $\phi_D : C \to \mathbb{P}^N_k$ associated to a divisor $D$ on a smooth projective curve $C$ — where $N = \ell(D) - 1$ and we use a basis $f_0, \ldots, f_N$ of $\mathcal{L}(D)$ — is an embedding if and only if $D$ satisfies condition $(*)$. The hypotheses of this theorem are: \begin{itemize} \item $C$ is a smooth projective curve and $D$ is a divisor on $C$ — given, with $D = 3K_C$. \item $f_0, \ldots, f_N$ is a basis for $\mathcal{L}(D)$ — fix any such basis; the map $\phi_D$ is well-defined up to a linear change of coordinates on $\mathbb{P}^N_k$, so the property of being an embedding is independent of the choice of basis. \end{itemize} By Step 3, the divisor $D = 3K_C$ satisfies condition $(*)$. Therefore $\phi_{3K_C} : C \to \mathbb{P}^N_k$ is an embedding. [/step] [step:Compute the dimension of the target as $N = 5g - 6$ via Riemann–Roch] It remains to identify the integer $N = \ell(3K_C) - 1$ as $5g - 6$, so that the embedding lands in $\mathbb{P}^{5g - 6}_k$ as claimed. We apply [Riemann–Roch for Large Degree Divisors](/theorems/2188), which states: if $D$ is a divisor on $C$ with $\deg D \geq 2g - 1$, then $\ell(K_C - D) = 0$ and \begin{align*} \ell(D) = \deg D - g + 1. \end{align*} We verify the hypothesis with $D = 3K_C$. By Step 1, $\deg(3K_C) = 6g - 6$. The required inequality is \begin{align*} 6g - 6 \geq 2g - 1 \quad \Longleftrightarrow \quad 4g \geq 5 \quad \Longleftrightarrow \quad g \geq 5/4. \end{align*} For integer $g \geq 2$ this is satisfied (in fact $4g - 5 \geq 3$ for $g \geq 2$). So the hypothesis holds, and \begin{align*} \ell(3 K_C) = \deg(3 K_C) - g + 1 = (6g - 6) - g + 1 = 5g - 5. \end{align*} Hence the linear-system map lands in projective space of dimension \begin{align*} N = \ell(3 K_C) - 1 = (5g - 5) - 1 = 5g - 6, \end{align*} matching the statement. Combined with Step 4, this gives an embedding \begin{align*} \phi_{3 K_C} : C \hookrightarrow \mathbb{P}^{5g - 6}_k. \end{align*} This completes the proof. [guided] The dimension $5g - 6$ is the unique value forced on us by Riemann–Roch — it is not a choice. Once we know $3K_C$ has degree $6g - 6$ and lies above the threshold $2g - 1$ where the Riemann–Roch correction term vanishes, $\ell(3K_C)$ is rigidly determined by the degree-genus formula \begin{align*} \ell(D) = \deg D - g + 1, \qquad \deg D \geq 2g - 1. \end{align*} Plug in $\deg D = 6g - 6$ to get $\ell(D) = 6g - 6 - g + 1 = 5g - 5$, and the projective dimension is one less, $5g - 6$. \textbf{Sanity check at $g = 2$.} A genus-$2$ curve has $\deg(3K_C) = 6$ and $\ell(3K_C) = 5g - 5 = 5$, so $\phi_{3K_C}$ embeds $C$ into $\mathbb{P}^4_k$. This matches the classical fact: every genus-$2$ curve is hyperelliptic (so the canonical $\phi_{K_C}$ is not an embedding — it is the degree-$2$ map to $\mathbb{P}^1_k$), but $3K_C$ saves us by yielding an embedding into $\mathbb{P}^4_k$. \textbf{What did we use? Why is the proof so short?} The hard work has been done in the prerequisite theorems. [Large Degree Implies Condition $(*)$](/theorems/2196) reduces an "embedding" question — which a priori involves derivatives of regular functions and a closed-immersion verification — to a numerical condition on a single divisor. [Riemann–Roch for Large Degree Divisors](/theorems/2188) then turns the numerical condition into elementary degree arithmetic. Once you have these two tools, the triple canonical embedding is a one-line consequence: "$6g - 6 > 2g$ for $g \geq 2$, done." [/guided] [/step]