^2 \bigl( L _f ^\bullet , {\mathcal O}_C \bigr)$ vanishes as well, i.e. the map is unobstructed, the space of stable maps has the expected dimension at $f$ and the point represented by $f$ is smooth (for the stack).

If we consider the dual sequence of (\[preserra\]) and use Serre duality we obtain the sequence $$\xymatrix @R=3pt {0 \ar[r] & \bigl( {\rm {\mathbb E}xt } ^2 \bigr) ^{\vee } \ar[r] & 
{\rm H}^0 (C, f^* \Omega ^1 _X \otimes \omega _C ) \ar[r] ^\alpha & 
{\rm H} ^0 (C , \Omega ^1 _C \otimes \omega _C ) \ar[r] & \\
\ar[r] & \bigl( {\rm {\mathbb E}xt } ^1 \bigr) ^{\vee } \ar[r] & 
{\rm H} ^1 (C, f^* \Omega ^1 _X \otimes \omega _C ) \ar[r] & 
{\rm H} ^1 (C, \Omega ^1 _C \otimes \omega _C) \ar[r] & 0}$$

It is easy to convince oneself that the morphism $\alpha $ is the morphism induced by the differential map $df : f^* \Omega ^1 _X \longrightarrow \Omega _C$, by tensoring with the dualizing sheaf and taking global sections. Associated to $f$ we may define the sheaves ${\mathcal C}_f$ and ${\mathcal Q}_f$ on $C$, by requiring the following sequence to be 