[proofplan]
We tensor the defining exact sequence of the first infinitesimal neighbourhood of $x$ by the line bundle $M$. Since $M$ is locally free of rank one, exactness is preserved, and the resulting long exact sequence in sheaf cohomology identifies the cokernel of the first-jet evaluation map with a subspace of $H^1(X,M\otimes \mathcal I_x^2)$. The assumed vanishing forces this cokernel to be zero. Finally, a local coordinate system and a local frame for $M$ identify the quotient by $\mathcal I_x^2$ with the constant and linear parts of a local section, which is precisely first-jet separation at $x$.
[/proofplan]
[step:Tensor the first infinitesimal neighbourhood sequence by the line bundle]
Let $\mathcal I_x \subset \mathcal O_X$ be the ideal sheaf of functions vanishing at $x$. The quotient sheaf $\mathcal O_X/\mathcal I_x^2$ is the structure sheaf of the first infinitesimal neighbourhood of $x$, and there is a short exact sequence of sheaves
\begin{align*}
0\longrightarrow \mathcal I_x^2\longrightarrow \mathcal O_X\longrightarrow \mathcal O_X/\mathcal I_x^2\longrightarrow 0.
\end{align*}
Because $M$ is a holomorphic line bundle, its sheaf of holomorphic sections is locally free of rank one over $\mathcal O_X$. Therefore tensoring with $M$ over $\mathcal O_X$ preserves exactness. We obtain the short exact sequence
\begin{align*}
0\longrightarrow M\otimes_{\mathcal O_X}\mathcal I_x^2\longrightarrow M\longrightarrow M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2\longrightarrow 0.
\end{align*}
Here the middle arrow sends a local section of $M$ to its class modulo sections whose local expression vanishes to order at least two at $x$.
[/step]
[step:Use the long exact cohomology sequence to prove surjectivity]
Let
\begin{align*}
\rho:H^0(X,M)\longrightarrow H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2)
\end{align*}
be the map induced on global sections by the quotient morphism
\begin{align*}
M\longrightarrow M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2.
\end{align*}
Applying sheaf cohomology to the short exact sequence from the previous step gives the exact segment
\begin{align*}
H^0(X,M)\xrightarrow{\rho}H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2)\xrightarrow{\delta}H^1(X,M\otimes_{\mathcal O_X}\mathcal I_x^2),
\end{align*}
where $\delta$ is the connecting homomorphism. Exactness at the middle term means
\begin{align*}
\operatorname{im}\rho=\ker\delta.
\end{align*}
By hypothesis,
\begin{align*}
H^1(X,M\otimes_{\mathcal O_X}\mathcal I_x^2)=0.
\end{align*}
Hence $\delta$ is the zero map into the zero [vector space](/page/Vector%20Space), so
\begin{align*}
\ker\delta=H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2).
\end{align*}
Therefore $\operatorname{im}\rho$ is the whole target, and $\rho$ is surjective.
[guided]
The goal is to show that every section of the quotient sheaf $M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2$ comes from an honest global section of $M$. The quotient map of sheaves
\begin{align*}
M\longrightarrow M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2
\end{align*}
induces the global-section map
\begin{align*}
\rho:H^0(X,M)\longrightarrow H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2).
\end{align*}
Surjectivity of $\rho$ is exactly the statement that every infinitesimal first-order datum at $x$ lifts to a global section.
We now use the short exact sequence
\begin{align*}
0\longrightarrow M\otimes_{\mathcal O_X}\mathcal I_x^2\longrightarrow M\longrightarrow M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2\longrightarrow 0.
\end{align*}
This sequence is exact because the sheaf of holomorphic sections of $M$ is locally free of rank one over $\mathcal O_X$, hence flat, so tensoring the defining sequence for $\mathcal O_X/\mathcal I_x^2$ by $M$ preserves exactness. The long exact sequence in sheaf cohomology gives the exact segment
\begin{align*}
H^0(X,M)\xrightarrow{\rho}H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2)\xrightarrow{\delta}H^1(X,M\otimes_{\mathcal O_X}\mathcal I_x^2),
\end{align*}
where $\delta$ is the connecting homomorphism. Exactness at the middle vector space says that an element of $H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2)$ lifts to $H^0(X,M)$ precisely when it is killed by $\delta$:
\begin{align*}
\operatorname{im}\rho=\ker\delta.
\end{align*}
The hypothesis is exactly the vanishing of the obstruction target:
\begin{align*}
H^1(X,M\otimes_{\mathcal O_X}\mathcal I_x^2)=0.
\end{align*}
Thus every element of $H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2)$ is automatically in $\ker\delta$, because $\delta$ maps into the zero vector space. Therefore
\begin{align*}
\operatorname{im}\rho=H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2).
\end{align*}
This proves that $\rho$ is surjective.
[/guided]
[/step]
[step:Identify the quotient with first-order jet data at $x$]
Let $n=\dim_{\mathbb C}X$. Choose a holomorphic coordinate chart $(U,z)$ centred at $x$, where $z:U\to z(U)\subset \mathbb C^n$ satisfies $z(x)=0$, and write $z_i:U\to\mathbb C$ for its coordinate functions. Choose a nowhere-vanishing holomorphic local frame $e$ of $M$ over $U$. Every local section $s$ of $M$ over $U$ is uniquely of the form
\begin{align*}
s=f e
\end{align*}
for a [holomorphic function](/page/Holomorphic%20Function) $f:U\to\mathbb C$.
Let $\mathcal O_{X,x}$ denote the stalk of $\mathcal O_X$ at $x$, and let $\mathcal I_{x,x}\subset \mathcal O_{X,x}$ denote the stalk of the ideal sheaf $\mathcal I_x$ at $x$. The stalk quotient $\mathcal O_{X,x}/\mathcal I_{x,x}^2$ records exactly the class of $f$ modulo germs of holomorphic functions vanishing to order at least two at $x$. In the chosen coordinates, this class is determined by
\begin{align*}
f(x)
\end{align*}
and by the first partial derivatives
\begin{align*}
\frac{\partial f}{\partial z_i}(x)\quad\text{for }1\le i\le n.
\end{align*}
Thus $H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2)$ is the space of possible first-order jets of local sections of $M$ at $x$, expressed in the frame $e$. Since the map $\rho$ is surjective, every prescribed first-order jet at $x$ is realised by a global section of $M$. Hence $M$ separates first jets at $x$, equivalently the linear system $|M|$ separates first jets at $x$.
[guided]
Let $n=\dim_{\mathbb C}X$. Choose a holomorphic coordinate chart $(U,z)$ centred at $x$, so $z:U\to z(U)\subset \mathbb C^n$ is a biholomorphism onto an open subset and $z(x)=0$. Write $z_i:U\to\mathbb C$ for the $i$-th coordinate function, for $1\le i\le n$. Choose a nowhere-vanishing holomorphic local frame $e$ of $M$ over $U$. Then every local section $s$ of $M$ over $U$ has a unique expression
\begin{align*}
s=f e
\end{align*}
for a holomorphic function $f:U\to\mathbb C$.
Let $\mathcal O_{X,x}$ denote the stalk of $\mathcal O_X$ at $x$, and let $\mathcal I_{x,x}\subset \mathcal O_{X,x}$ denote the stalk of the ideal sheaf $\mathcal I_x$ at $x$. The quotient $\mathcal O_{X,x}/\mathcal I_{x,x}^2$ forgets all terms of order at least two in the local Taylor expansion at $x$. In the coordinates $z_1,\dots,z_n$, the class of $f$ modulo $\mathcal I_{x,x}^2$ is therefore determined exactly by its constant term
\begin{align*}
f(x)
\end{align*}
and by its first partial derivatives
\begin{align*}
\frac{\partial f}{\partial z_i}(x)\quad\text{for }1\le i\le n.
\end{align*}
The sheaf $M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2$ is supported at the single point $x$, so its global sections are the same first-order data at that stalk. Under the frame $e$, an element of
\begin{align*}
H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2)
\end{align*}
is precisely a prescribed value and prescribed first partial derivatives for the local expression of a section of $M$ at $x$.
The previous step proved that the map
\begin{align*}
\rho:H^0(X,M)\longrightarrow H^0(X,M\otimes_{\mathcal O_X}\mathcal O_X/\mathcal I_x^2)
\end{align*}
is surjective. Hence every such first-order datum is the image of a global holomorphic section of $M$. This is exactly first-jet separation at $x$, and the same property is described by saying that the complete linear system $|M|$ separates first jets at $x$.
[/guided]
[/step]
