Let $X$ be a smooth curve over $\mathbb{C}$. I wish to compute the deformation of the following data $(E,F,x)$. $E$ and $F$ are locally free sheaves over $X$ and $x$ is a point on $X$. They satisfy:


1) $E\longrightarrow F$ is a injective map of sheaves over $X$.


2) $F/E$ has length 1 and is supported at $x$. 


How to compute the first order deformation of $(E,F,x)$ and describe it as a subspace of $H^{1}(End(E))\oplus H^{1}(End(F)) \oplus T\_{x}X?$



Deformation of "Hecke modification"
A morphism $F \to E$ can be thought of as a representation of the quiver 
$$
A\_2 = \{ \bullet \to \bullet \}
$$
in the category $Coh(X)$. The category $Rep(A\_2,Coh(X))$ is abelian, so what you need is $Ext^1$ in this category. If $S\_1$ and $S\_2$ denote the simple modules of the first and the second vertices of the quiver, there is an exact sequence
$$
0 \to E \otimes S\_2 \to C \to F \otimes S\_1 \to 0,\qquad(\*)
$$
where $C = \{F \to E\}$. Using the fact that $Ext^\bullet(S\_i,S\_i) = \Bbbk$, 
$Ext^\bullet(S\_1,S\_2) = \Bbbk[-1]$, $Ext^\bullet(S\_2,S\_1) = 0$, one gets a long exact sequence
$$
0 \to Hom(C,C) \to Hom(E,E) \oplus Hom(F,F) \to Hom(F,E) \to Ext^1(C,C) \to Ext^1(E,E) \oplus Ext^1(F,F) \to Ext^1(F,E) \to \dots
$$
And the space $Ext^1(C,C)$ in its middle is the tangent space to deformations.


EDIT. To get the long exact sequence, note first that applying to $(\*)$ the functor $Hom(-,E\otimes S\_2)$ one gets
$$
0 \to 0 \to Hom(C,E\otimes S\_2) \to Hom(E,E) \to Hom(F,E) \to Ext^1(C,E\otimes S\_2) \to Ext^1(E,E) \to Ext^1(F,E) \to \dots
$$
and applying to $(\*)$ the functor $Hom(-,F \otimes S\_1)$ one gets
$$
Ext^i(C,F \otimes S\_1) \cong Ext^i(F,F).
$$
Now, finally, applying to $(\*)$ the functor $Hom(C,-)$ and using the above identifications, one gets the desired sequence.


