contains at least three singular points of $C$ and every contracted component of geometric genus one contains at least one.

We are interested in computing the obstruction space to deforming the stable map $f : C \rightarrow X$. Let $f^* \Omega ^1 _X \rightarrow
\Omega ^1 _C$ be the natural complex of sheaves associated with the differential of $f$ and where the sheaf $f^* \Omega ^1 _X$ is in degree -1 and the sheaf $\Omega ^1 _C$ is in degree 0. We know that the stability condition is equivalent to the vanishing of the group ${\rm Hom} \bigl( f^* \Omega ^1 _X \rightarrow
\Omega ^1 _C , {\mathcal O}_C \bigr)$. The tangent space to $\overline {\mathcal M}_{0,0} \bigl( X, \beta \bigr)$ at $f$ is given by the hypercohomology group ${\rm {\mathbb E}xt } ^1 \bigl( f^* \Omega ^1 _X \rightarrow
\Omega ^1 _C , {\mathcal O}_C \bigr)$. The obstruction space is a quotient of the hypercohomology group ${\rm {\mathbb E}xt } ^2 \bigl( f^* \Omega ^1 _X \rightarrow \Omega ^1 _C , 
{\mathcal O}_C \bigr)$. Denote by $L _f ^\bullet $ the complex $f^* \Omega ^1 _X \rightarrow \Omega ^1 _C$, where the first sheaf is in degree -1 and 