My question is related to [this question][1], but I'm looking for something a bit more explicit.

Let $S$ be a smooth surface over $\mathbb C$, fix a point $s\in S$ and take the blow up $\beta \colon X \rightarrow S$ at $s$. Let $E$ be the exceptional divisor.

Then the tangent sheaves of $S$ and $X$ are related by the exact sequence 
$$
0 \rightarrow T_X \rightarrow \beta^*T_S \rightarrow {\mathcal N}_{E|X} \rightarrow 0
$$
where ${\mathcal N}_{E|X}$ is the normal bundle of $E$ in $X$. I'm interested in writing the induced map
$$
\partial \colon H^0({\mathcal N}_{E|X} ) \rightarrow H^1(T_X)
$$
explicitely in the following sense. 

Choose an affine chart $U$ centered at $s$ inducing local coordinates $x,y$: it naturally induces homogeneous coordinates $(X,Y)$ on $E\cong {\mathbb P}^1$ by writing a neighbourhood of $E$ as the locus  $xY=yX$ on $U \times {\mathbb P}^1$. Since ${\mathcal N}_{E|X}\cong {\mathcal O}_{\mathbb P^1}(1)$ I may now write
$$
H^0({\mathcal N}_{E|X} )=\left\{ aX+bY | a,b \in {\mathbb C}\right\}
$$
On the other hand by the wellknown work of M. Schlessinger, setting for simplicity ${\mathcal C}:=\mathbf{Spec} \left( {\mathbb C}[t]/t^2 \right)$ we have a set-theoretic identification
$$
H^1(T_X)=\left\{ \text{flat families } {\mathcal X} \rightarrow {\mathcal C} | {\mathcal X} \times_{\mathcal C} {\mathbb C}\cong X\right\}/isomorphisms
$$
I would like to construct the family image of a chosen linear form $aX+bY$. 

By the natural interpretation of the map $\partial$ ("move the point $s$") it seems to me that it should be the blow-up of $S \times {\mathcal C}$ on a subscheme of $U \times {\mathcal C}$ of the form $x-ct=y-dt=0$ where $c,d \in {\mathbb C}$ depends in some easy way from $a$ and $b$ but I can't prove it.

I'm rather sure this is an easy exercise for experts but I cannot find it written anywhere, a reference is most than welcome. 



  [1]: https://mathoverflow.net/questions/198049/deformations-of-a-blowup