My question is related to this question, 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 $\beta \colon S' \rightarrow S$ at $s$. Let $E$ be the exceptional divisor.

Then the tangent sheaves of $S$ and $S'$ are related by the exact sequence $$ 0 \rightarrow T_{S'} \rightarrow \beta^*T_S \rightarrow {\mathcal O}_{E} (-E) \rightarrow 0 $$ I'm interested in writing the induced map $$ \partial \colon H^0({\mathcal O}_{E} (-E) ) \rightarrow H^1(T_{S'}) $$ 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 ${\mathcal O}_{E} (-E) \cong {\mathcal O}_{\mathbb P^1}(1)$ I may now write $$ H^0({\mathcal O}_{E} (-E) )=\left\{ aX+bY | a,b \in {\mathbb C}\right\} $$ 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 $$ H^1(T_{S'})=\left\{ \text{flat families } {\mathcal X} \rightarrow {\mathcal C} | {\mathcal X} \times_{\mathcal C} {\mathbb C}\cong S'\right\}/isomorphisms $$ 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.