In the Huybrecht's book Fourier Mukai transform in Algebraic geometry, in order to prove Bondal and Orlov's results (proposition 4.11), it seems that he uses following things.
Suppose $k$ is an algebraically close field, $X$ is a smooth projective variety over $k$. $L$ is a line bundle on $X$ and $x$ is a closed point $X$. $k(x)\cong k$ is the residue field and $i: x\hookrightarrow X$ is the closed embedding.
Then he claims two things:
1: A closed subscheme $Z_{x}$ of $X$ with length 2 concentrated in $x\in X$ is one to one correponds a nontrival element of $\mathrm{Ext}^{1}(i_{*}(k(x)),i_{*}(k(x)))$.
2: $L$ separated the tangent vectors at $x$ if and only if $H^{0}(X,L)\to H^{0}(X, \mathcal{O}_{Z_{x}})$ induced by the restriction map is surjecitive for all $Z_{x}$ (closed subscheme of length 2 concentrated in $x$).
First, I do not know what is the restriction map $L\to \mathcal{O}_{Z_{x}}$ in the claim 2. I think the first claim just means a tangent vector at $x$ is one to one correponds a closed subscheme of length 2 concentrated in $x$. Could you give me some ideas how to prove these two claims?