Let $X$ be a compact complex manifold, let $L \to X$ be a holomorphic line bundle, and let $s \in H^0(X,L)$ be a holomorphic section whose zero locus $D=(s)_0$ is a smooth divisor. Let

\begin{align*} \rho:H^0(X,L)\to H^0(D,N_{D/X}) \end{align*}

be the map obtained by restricting sections to $D$ and then using the divisor isomorphism $L|_D\cong N_{D/X}$. Regard the complete linear system $|L|$ as the projective space of nonzero sections of $L$ modulo scalar multiplication, and regard a first-order motion of $D$ in $|L|$ through the divisor $(s)_0$ as a first-order section $s+\varepsilon t$ over $\operatorname{Spec}\mathbb C[\varepsilon]/(\varepsilon^2)$, with $t\in H^0(X,L)$, modulo replacing $t$ by $t+c s$ for $c\in\mathbb C$.

Then $\operatorname{im}\rho$ is exactly the subspace of $H^0(D,N_{D/X})$ consisting of embedded first-order deformation classes of $D$ in $X$ obtained by such first-order motions in $|L|$. Equivalently, for every $t\in H^0(X,L)$, the first-order family of divisors cut out by $s+\varepsilon t$ has embedded deformation class $\rho(t)$, and every embedded first-order deformation coming from a first-order motion in $|L|$ is obtained in this way.