[proofplan] A morphism $\phi: C \to \mathbb{P}^N_k$ from a smooth projective curve is a closed embedding if and only if (i) it is injective on closed points, and (ii) it induces an injection on tangent spaces at every closed point — equivalently, $\phi$ is "injective on points and on first-order infinitesimal neighbourhoods". The two halves of condition $(*)$ correspond exactly to these two requirements: (i) the case $p \neq q$ in condition $(*)$ — that some function in $\mathcal{L}(D)$ separates $p$ from $q$ — gives injectivity at the level of closed points, and (ii) the case $p = q$ in condition $(*)$ — that some function in $\mathcal{L}(D)$ has a simple zero at $p$ relative to others vanishing at $p$ — gives injectivity on tangent spaces. The proof translates each clause of condition $(*)$ into a sub-isomorphism property of $\phi_D$ and combines them via the standard closed-immersion criterion for morphisms to projective space. [/proofplan]

[step:Reduce to base-point-freeness, injectivity on points, and injectivity on tangent spaces] A morphism $\phi: C \to \mathbb{P}^N_k$ from a smooth projective curve $C$ over an algebraically closed field $k$ is a closed embedding if and only if all of the following hold: 1. $\phi$ is everywhere defined as a morphism (no base points). 2. $\phi$ is injective on closed points: for $p \neq q$ in $C(k)$, $\phi(p) \neq \phi(q)$. 3. The induced map on tangent spaces $d\phi_p: T_p C \to T_{\phi(p)} \mathbb{P}^N_k$ is injective for every closed point $p \in C$. This is the standard closed-immersion criterion: condition (1) makes $\phi$ a morphism, conditions (2) and (3) make the morphism set-theoretically and scheme-theoretically injective; together with properness of $C$ (since $C$ is projective), these conditions imply $\phi$ is a closed immersion. [/step]

[step:Translate condition $(*)$ for $p \neq q$ into base-point-freeness and injectivity on points] Fix two distinct closed points $p \neq q$ in $C$. Recall the base-point and separation chain of inequalities, valid for any divisor $D$ on a smooth projective curve and any closed point $r \in C$: \begin{align*} \ell(D) - 1 \leq \ell(D - r) \leq \ell(D), \end{align*} because $\mathcal{L}(D - r) \subseteq \mathcal{L}(D)$ is the subspace of functions in $\mathcal{L}(D)$ vanishing at $r$, of codimension at most $1$. [claim:Condition $(*)$ at $\{p, q\}$ for $p \neq q$ implies $\ell(D - p) = \ell(D) - 1$ and $\ell(D - p - q) = \ell(D - p) - 1$] Suppose $\ell(D - p - q) = \ell(D) - 2$. Combining with $\ell(D) - 1 \leq \ell(D - p) \leq \ell(D)$ and $\ell(D - p) - 1 \leq \ell(D - p - q) \leq \ell(D - p)$: \begin{align*} \ell(D) - 2 = \ell(D - p - q) \geq \ell(D - p) - 1 \geq (\ell(D) - 1) - 1 = \ell(D) - 2, \end{align*} forcing equality throughout: $\ell(D - p) = \ell(D) - 1$ and $\ell(D - p - q) = \ell(D - p) - 1$. [/claim] [proof] The two inequalities used are $\ell(D) - 1 \leq \ell(D - p)$ and $\ell(D - p) - 1 \leq \ell(D - p - q)$, both standard. The chain forces both inequalities to be equalities, giving the claimed values. [/proof] The two conclusions of the claim translate into geometric properties of $\phi_D$: - **Base-point-freeness at $p$:** $\ell(D - p) = \ell(D) - 1$ means the linear functional "evaluation at $p$" $\operatorname{ev}_p: \mathcal{L}(D) \to k$ — well-defined after dividing by a uniformiser — is surjective onto $k$, so its kernel $\mathcal{L}(D - p)$ is of codimension $1$. Equivalently, there exists $f \in \mathcal{L}(D)$ with $f(p) \neq 0$ (after the standard rescaling; concretely, after choosing local coordinates trivialising the line bundle $\mathcal{O}(D)$ near $p$, this is the statement that the map $\phi_D$ is well-defined at $p$). This means $p$ is not a base point of the linear system, and therefore $\phi_D$ is defined at $p$ without singularity. - **Separation of $p$ from $q$:** $\ell(D - p - q) = \ell(D - p) - 1$ means there exists $f \in \mathcal{L}(D - p)$ with $f(q) \neq 0$. Equivalently, there exists $f \in \mathcal{L}(D)$ vanishing at $p$ but not at $q$. The image points $\phi_D(p)$ and $\phi_D(q)$ in $\mathbb{P}^N_k$ are then distinguished by the homogeneous coordinate $f$: $\phi_D(p)$ has $f$-coordinate $0$ while $\phi_D(q)$ has $f$-coordinate $\neq 0$. Hence $\phi_D(p) \neq \phi_D(q)$ in $\mathbb{P}^N_k$. Conversely, if base-point-freeness at $p$ and separation of $p, q$ both hold, the same chain of inequalities forces $\ell(D - p - q) = \ell(D) - 2$, recovering condition $(*)$ at $\{p, q\}$. Applying this for all pairs $p \neq q \in C$: condition $(*)$ on all pairs $\{p, q\}$ with $p \neq q$ holds if and only if $\phi_D$ is base-point-free and injective on closed points. [/step]

[step:Translate condition $(*)$ for $p = q$ into injectivity on tangent spaces] Now fix a single closed point $p \in C$. The case $p = q$ of condition $(*)$ reads \begin{align*} \ell(D - 2p) = \ell(D) - 2. \end{align*} We use the chain \begin{align*} \ell(D - 2p) \geq \ell(D - p) - 1 \geq \ell(D) - 2, \end{align*} which by the same forcing argument as in the claim implies $\ell(D - 2p) = \ell(D - p) - 1 = \ell(D) - 2$ when condition $(*)$ at $\{p, p\}$ holds. The geometric translation is: - **Base-point-freeness at $p$ (already obtained from Step 2):** $\ell(D - p) = \ell(D) - 1$. - **Injectivity of the differential at $p$:** $\ell(D - 2p) = \ell(D - p) - 1$ means there exists $f \in \mathcal{L}(D - p)$ with $f \notin \mathcal{L}(D - 2p)$ — equivalently, $f$ has order exactly $1$ at $p$ as a section of $\mathcal{O}(D)$ (vanishes at $p$ but not to second order). Choose a local uniformiser $t$ at $p$. After choosing a local trivialisation of $\mathcal{O}(D)$ near $p$, the morphism $\phi_D$ is locally given by $p \mapsto [f_0(p) : \cdots : f_N(p)]$, and the differential $d(\phi_D)_p$ is determined by the leading-order vanishing behaviour of $f_0, \ldots, f_N$ at $p$. The existence of $f \in \mathcal{L}(D - p)$ with $\operatorname{ord}_p(f) = 1$ — that is, $f$ has order exactly $1$ at $p$ — shows that, after passing to a basis adapted to the filtration $\mathcal{L}(D) \supset \mathcal{L}(D - p) \supset \mathcal{L}(D - 2p)$, one of the basis elements has nonzero linear term at $p$, so the differential $d(\phi_D)_p$ has a nonzero entry — equivalently, has rank $\geq 1$. Since $\dim T_p C = 1$, the differential $d(\phi_D)_p: T_p C \to T_{\phi_D(p)} \mathbb{P}^N_k$ is injective. Conversely, if $\phi_D$ is base-point-free at $p$ and has injective differential $d(\phi_D)_p$, then by reading the same local computation backwards, $\mathcal{L}(D - p)$ contains a function of order exactly $1$ at $p$, so $\ell(D - 2p) = \ell(D - p) - 1$, recovering condition $(*)$ at $\{p, p\}$. [/step]

[step:Combine the two cases to conclude]Combining Steps 2 and 3: \begin{align*} \text{$D$ satisfies condition $(*)$ for all pairs $\{p, q\}$, $p \neq q$} &\iff \text{$\phi_D$ base-point-free and injective on points,} \\ \text{$D$ satisfies condition $(*)$ for all pairs $\{p, p\}$} &\iff \text{$\phi_D$ base-point-free with injective differential everywhere}. \end{align*} Combined: $D$ satisfies condition $(*)$ for all $p, q \in C$ if and only if $\phi_D$ is base-point-free, injective on closed points, and has injective differential everywhere — exactly the closed-immersion criterion of Step 1. Hence $D$ satisfies condition $(*)$ if and only if $\phi_D: C \to \mathbb{P}^N_k$ is a closed embedding.[/step]

[guided]The proof rests on a single observation: condition $(*)$ — the equality $\ell(D - p - q) = \ell(D) - 2$ — encodes *exactly two units of separation* by functions in $\mathcal{L}(D)$ at each pair of points, and these two units correspond to the two geometric requirements for an embedding. **The two universal inequalities.** For any divisor $D$ on a smooth projective curve $C$ and any closed point $r$, the inclusion $\mathcal{L}(D - r) \subseteq \mathcal{L}(D)$ has codimension at most $1$ — the codimension is $1$ if some function in $\mathcal{L}(D)$ does not vanish at $r$ (so $r$ is not a base point), and $0$ if every function in $\mathcal{L}(D)$ vanishes at $r$ (so $r$ is a base point). Iterating gives \begin{align*} \ell(D) - 2 \leq \ell(D - p - q) \leq \ell(D), \end{align*} with strict inequality at most $2$. **The forcing structure.** Condition $(*)$ requires the *minimum* drop $\ell(D) - 2$. By the chain $\ell(D - p - q) \geq \ell(D - p) - 1 \geq \ell(D) - 2$, the minimum is achieved only if both inequalities are equalities — meaning both intermediate codimensions are exactly $1$. This forces base-point-freeness at $p$ (first inequality) AND separation of $p$ from $q$ by some function vanishing at $p$ (second inequality). When $p = q$, the second inequality becomes "the function in $\mathcal{L}(D - p)$ has order exactly $1$ at $p$", i.e. tangent-space injectivity. **Why "exactly two" is the right number.** Geometrically, $\phi_D$ embeds $C$ into $\mathbb{P}^N_k$ if it injects the closed points (one unit of separation per pair $p \neq q$) and the tangent vectors at each point (one more unit at $p = q$). Each unit of separation costs one dimension in $\mathcal{L}(D)$. So an embedding requires "two units of $\mathcal{L}(D)$ to be expended" at every pair $\{p, q\}$ — exactly the content of condition $(*)$. **Why the converse direction works.** The hard part of the converse is interpreting the equality $\ell(D - p) = \ell(D) - 1$ and $\ell(D - p - q) = \ell(D - p) - 1$ as geometric properties of $\phi_D$. The first equality says "evaluation at $p$" $\mathcal{L}(D) \to k$ is surjective, equivalently, $p$ is not a base point and $\phi_D$ is defined at $p$. The second says "evaluation at $q$" $\mathcal{L}(D - p) \to k$ is also surjective, equivalently, there is a function in $\mathcal{L}(D)$ vanishing at $p$ but not at $q$ — separating $\phi_D(p)$ from $\phi_D(q)$. The case $p = q$ replaces "evaluation at $q$" by "first-order Taylor coefficient at $p$", giving tangent-space injectivity. **The closed-immersion criterion in projective space.** The theorem we use in Step 1 — that a morphism from a smooth projective curve to $\mathbb{P}^N_k$ is a closed embedding iff it is injective on closed points and on tangent spaces — is a special case of the general criterion: a morphism of finite type between schemes over an algebraically closed field is a closed immersion iff it is proper (automatic for $C$ projective), injective on closed points, and injective on tangent spaces (equivalently, formally unramified). The smooth-curve hypothesis on $C$ guarantees that the tangent-space injectivity is exactly the differential-injectivity condition; for non-smooth $C$, one needs the formal-cotangent version of the criterion. **The converse argument in Step 3 (tangent-space injectivity).** Locally near $p$, after a choice of uniformiser $t$ at $p$ and a local trivialisation of $\mathcal{O}_C(D)$, each $f_i \in \mathcal{L}(D)$ can be written as a Laurent expansion in $t$ near $p$. Adapting the basis $\{f_0, \ldots, f_N\}$ to the filtration $\mathcal{L}(D) \supset \mathcal{L}(D - p) \supset \mathcal{L}(D - 2p)$ — say by Gauss elimination — we may assume $f_0$ has order $0$ at $p$ (using $\ell(D) - \ell(D - p) = 1$, base-point-freeness), $f_1$ has order exactly $1$ at $p$ (using $\ell(D - p) - \ell(D - 2p) = 1$), and the remaining $f_2, \ldots, f_N$ all have order $\geq 2$. In affine coordinates near $p$, divide by $f_0$: \begin{align*} \phi_D(t) = \left[1 : \frac{f_1(t)}{f_0(t)} : \frac{f_2(t)}{f_0(t)} : \cdots : \frac{f_N(t)}{f_0(t)}\right]. \end{align*} The $f_1$-coordinate behaves like $c_1 t + O(t^2)$ for some $c_1 \neq 0$ (since $f_1$ has order $1$ at $p$ and $f_0$ does not vanish), so the differential is nonzero in the $f_1$-direction — proving $d(\phi_D)_p$ is injective. **A summary table.** | Condition $(*)$ at $\{p, q\}$ | Geometric content | |---|---| | $p \neq q$, base-point-freeness at $p$ | $\phi_D$ defined at $p$ | | $p \neq q$, separation step | $\phi_D(p) \neq \phi_D(q)$ | | $p = q$, base-point-freeness | $\phi_D$ defined at $p$ | | $p = q$, tangent step | $d(\phi_D)_p$ injective |[/guided]