Let $X$ be a complex manifold of complex dimension $n$, let $M \to X$ be a holomorphic line bundle, and write $M$ also for its sheaf of holomorphic sections. Let $x \in X$, let $|M|$ denote the complete linear system associated to $M$, and let $\mathcal I_x \subset \mathcal O_X$ denote the ideal sheaf of holomorphic functions vanishing at $x$. If

\begin{align*} H^1(X,M\otimes_{\mathcal O_X}\mathcal I_x^2)=0, \end{align*}

then $M$, equivalently $|M|$, separates first jets at $x$: the first-jet evaluation map

\begin{align*} j_x^1:H^0(X,M)\longrightarrow H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2) \end{align*}

induced by the quotient morphism $\mathcal O_X \to \mathcal O_X/\mathcal I_x^2$ is surjective. Equivalently, after choosing any holomorphic coordinate chart centered at $x$ and any holomorphic local frame of $M$ near $x$, every prescribed constant term and $n$ first partial derivatives of the local expression of a section of $M$ at $x$ is realised by a global holomorphic section of $M$.