If we have the family of complex curves $f:X\rightarrow Y$, over a complex smooth curve $Y$ , we consider a fiber $C=f^{-1}(y)$ and its tangent bundle $T\_{C}$. We know that $df: f^{\*}{T\_{Y}}\_{|C}\cong N\_{C}.$ In fact both of these budnles are trivial and isomorphic to $C\times T\_{y}$(Voisin pg.223). Can we give explicitely this isomorphism, i.e if we take some vector from $N\_{C}$ how do we see it as an element of $f^{\*}{T\_{Y}}\_{|C}?$



Normal bundle of a fiber of the family of curves
Take a vector in $TX$ which represents the vector in $N\_C$; it is unique up to adding something tangent to $C$. Now apply $f'(x)$ to it, a linear map to $T\_y Y$. At a generic point of $X$, the map $f$ is a linear projection in local holomorphic coordinates, $f(x,y)=y$ and the map $f'$ is just $f$ in those coordinates. The fibers are $y=y\_0$ constant, the tangent to the fibers is $\partial\_x$, the normal bundle is spanned by $\partial\_{y}$ mod $\partial\_x$. Each element is linearly mapped by dropping the words "mod $\partial\_x$", i.e. $f\_\* \partial\_{y} \operatorname{mod} \partial\_x=\partial\_{y}$. 


