[proofplan] We compute the first-order motion locally in a trivialization of $L$. If $s=f e$ and $t=g e$, then the infinitesimal divisor is cut out by $f+\varepsilon g$, and its normal displacement along $D$ is the class of $g$ modulo the ideal $(f)$. Under the canonical divisor identification $L|_D\cong N_{D/X}$, these local classes are exactly the restriction of $t$ to $D$, and compatibility under changes of frame makes the construction global. Finally, first-order motions inside the complete linear system are precisely first-order variations $s+\varepsilon t$, modulo the projective rescaling that leaves the divisor unchanged, so their deformation classes are exactly the image of the restriction map. [/proofplan]

[step:Represent a first-order motion in the linear system by a varied section] Let $A=\mathbb C[\varepsilon]/(\varepsilon^2)$ be the ring of dual numbers. Let $X_A:=X\times \operatorname{Spec}A$, and let $L_A\to X_A$ denote the pullback of $L$. Since $A$ is the two-dimensional $\mathbb C$-[vector space](/page/Vector%20Space) $\mathbb C\oplus \varepsilon\mathbb C$ and $L_A$ is obtained from $L$ by extension of scalars, the space of global sections is $H^0(X_A,L_A)=H^0(X,L)\otimes_{\mathbb C}A$. Therefore an $A$-valued section reducing to $s$ modulo $\varepsilon$ has the unique form $s_A=s\otimes 1+t\otimes \varepsilon$, which we write as $s_A=s+\varepsilon t$, for a uniquely determined $t\in H^0(X,L)$. A first-order motion of the divisor $D$ inside the complete linear system $|L|$ is represented, after choosing a representative before projectivisation, by such a section. The corresponding first-order divisor is the Cartier divisor $D_A\subset X_A$ cut out by $s_A$. If $t$ is replaced by $t+c s$ for some $c\in \mathbb C$, then $s+\varepsilon(t+c s)=(1+c\varepsilon)s+\varepsilon t$. The element $1+c\varepsilon\in A^\times$ is a unit, so multiplication by it does not change the Cartier divisor. This is the infinitesimal form of projectivising the section space. [/step]

[step:Compute the local normal displacement of the divisor]Fix a point $p\in D$. Choose an open neighbourhood $U\subset X$ of $p$ and a holomorphic frame $e:\mathcal O_U\to L|_U$ for the line bundle $L$ over $U$. Define holomorphic functions $f:U\to \mathbb C$ and $g:U\to \mathbb C$ by the equations $s|_U=f e$ and $t|_U=g e$. Since $D$ is the smooth zero divisor of $s$, the ideal sheaf $\mathcal I_D|_U$ is generated by $f$, and $D\cap U$ is the smooth hypersurface $\{f=0\}$. Over $U_A:=U\times \operatorname{Spec}A$, the infinitesimal divisor associated to $s+\varepsilon t$ is cut out by $f+\varepsilon g\in \mathcal O_U\otimes_{\mathbb C}A$. The hypotheses of [citetheorem:9122] apply: $D\subset X$ is compact because $X$ is compact and $D$ is closed, and $D$ is a smooth complex submanifold because it is a smooth divisor. The divisor $D_A\subset X_A$ is an embedded first-order deformation over $A$ because it is a Cartier divisor whose reduction modulo $\varepsilon$ is the original embedded divisor $D\subset X$. Let $\mathcal I_D\subset \mathcal O_X$ be the ideal sheaf of $D$. On $U$, $\mathcal I_D|_U=(f)$, so the conormal sheaf is generated by $f\ \operatorname{mod}(f^2)$, and $N_{D/X}|_{D\cap U}=\mathcal Hom_{\mathcal O_D}(\mathcal I_D/\mathcal I_D^2,\mathcal O_D)|_{D\cap U}$. With the local convention used in the embedded-deformation identification, the first-order equation $f+\varepsilon g$ determines the homomorphism $\mathcal I_D/\mathcal I_D^2|_{D\cap U}\to \mathcal O_D|_{D\cap U}$ that sends $f\ \operatorname{mod}(f^2)$ to $g\ \operatorname{mod}(f)$. Hence on $D\cap U$ the embedded deformation class is represented by $g\ \operatorname{mod}(f)\in \mathcal O_D(D\cap U)$.[/step]

[guided]Let us unpack why the coefficient $g$ is the normal velocity and why [citetheorem:9122] applies. The theorem identifies embedded first-order deformations of a compact complex submanifold with sections of its normal bundle. Here $D\subset X$ is compact because $X$ is compact and $D$ is closed, and $D$ is a smooth complex submanifold because the zero divisor of $s$ is assumed smooth. The family $D_A\subset X_A$ is embedded in the fixed ambient space $X_A$ and is cut out locally by one equation reducing modulo $\varepsilon$ to the defining equation of $D$, so it is an embedded first-order deformation over the dual numbers. In the chosen frame $e$, the original divisor is cut out on $U$ by $f=0$, and the varied divisor is cut out on $U_A$ by $f+\varepsilon g=0$. Because $D$ is a smooth hypersurface, the ideal sheaf $\mathcal I_D|_U$ is the principal ideal $(f)$. Its conormal sheaf on $D\cap U$ is therefore generated by the class $f\ \operatorname{mod}(f^2)$, and the normal bundle is the dual sheaf $N_{D/X}|_{D\cap U}=\mathcal Hom_{\mathcal O_D}(\mathcal I_D/\mathcal I_D^2,\mathcal O_D)|_{D\cap U}$. The embedded first-order deformation records how a local defining equation changes modulo the original ideal. Here the defining equation changes from $f$ to $f+\varepsilon g$, so the coefficient of $\varepsilon$ defines the normal homomorphism $\mathcal I_D/\mathcal I_D^2|_{D\cap U}\to \mathcal O_D|_{D\cap U}$ that sends the generator $f\ \operatorname{mod}(f^2)$ to $g\ \operatorname{mod}(f)$. Thus the associated local section of $N_{D/X}$ is represented by $g\ \operatorname{mod}(f)\in \mathcal O_U/(f)=\mathcal O_D(D\cap U)$. This is the local form of the deformation class supplied by [citetheorem:9122].[/guided]

[step:Identify the local displacement with the restricted section of $L$] The divisor identification $L|_D\cong N_{D/X}$ for a smooth divisor is [citetheorem:7046]. In the frame $e$, the section $s|_U=f e$ identifies the local generator $e|_{D\cap U}$ of $L|_D$ with the normal homomorphism $\mathcal I_D/\mathcal I_D^2|_{D\cap U}\to \mathcal O_D|_{D\cap U}$ that sends $f\ \operatorname{mod}(f^2)$ to $1$. Since $t|_U=g e$, the restricted section $t|_D$ is represented by $g|_{D\cap U}=g\ \operatorname{mod}(f)$, and its image under $L|_D\cong N_{D/X}$ sends $f\ \operatorname{mod}(f^2)$ to $g\ \operatorname{mod}(f)$. Therefore the local normal displacement computed above is exactly the image of $t|_D$ under $L|_D\cong N_{D/X}$. This description is independent of the chosen frame. Indeed, if $e'=u e$ is another holomorphic frame on an [open set](/page/Open%20Set) $U'\subset U$, where $u:U'\to \mathbb C^\times$ is a nowhere-vanishing [holomorphic function](/page/Holomorphic%20Function), then $s=f'e'$ and $t=g'e'$ with $f'=u^{-1}f$ and $g'=u^{-1}g$. The local representative of the restricted section changes by the same transition function as the frame of $L|_D$, and the normal-bundle identification changes by the corresponding transition function. Hence the local classes glue to the global section $\rho(t)\in H^0(D,N_{D/X})$. [/step]

[step:Show the image is exactly the deformation space coming from the complete linear system] For every $t\in H^0(X,L)$, the divisor over $\operatorname{Spec}A$ cut out by $s+\varepsilon t$ is a first-order motion of $D$ inside $|L|$, and the previous steps show that its embedded deformation class is $\rho(t)$. Hence every such motion contributes an element of $\operatorname{im}\rho$. Conversely, by definition, an embedded first-order deformation of $D$ obtained by moving inside the complete linear system is represented by a first-order family of defining sections reducing to $s$ modulo $\varepsilon$. Such a family is of the form $s+\varepsilon t$ for some $t\in H^0(X,L)$, and its deformation class is $\rho(t)$ by the local computation above. Therefore the set of deformation classes arising from first-order motions in $|L|$ is contained in $\operatorname{im}\rho$. The two inclusions prove that this space of embedded first-order deformation classes is exactly $\operatorname{im}\rho\subset H^0(D,N_{D/X})$. [/step]