Let $C$ be a reduced, connected, projective and purely one-dimensional scheme of finite type over a field $k$.
Suppose that $C$ is rational, i.e. that the normalisation of $C$ is a disjoint union of copies of $\mathbb{P}^1\_k$.
Let $T\_C = \mathcal{H}om(\Omega^1\_C,\mathcal{O}\_C)$ be the tangent sheaf of $C$.


**Question:** Is it true that $H^1(C,T\_C)$ vanishes?


You may additionally assume that $C$ has only planar singularities, but I'm not sure this is needed.
The question is true if $C$ is smooth.
For context, a positive answer would imply that $C$ has no locally trivial deformations.



First cohomology of tangent sheaf of rational curve
Let $C$ be the union of 5 lines in general position in $\mathbb{P}^2$ (hence with 10 pairwise intersection points $P\_{ij}$, $1 \le i < j \le 5$) and let $F$ be the equation of $C$. We have the standard exact sequence
$$
0 \to \mathcal{O}\_C(-5) \stackrel{dF}\to \Omega\_{\mathbb{P}^2}\vert\_C \to \Omega\_C \to 0.
$$
Taking its dual we obtain an exact sequence
$$
0 \to T\_C \to T\_{\mathbb{P}^2}\vert\_C \stackrel{dF}\to \mathcal{O}\_C(5) \to \bigoplus\_{1 \le i < j \le 5} \mathcal{O}\_{P\_{ij}} \to 0.
$$
Now an easy computation of Euler characteristics gives
$$
\chi(T\_{\mathbb{P}^2}\vert\_C) = 5,
\quad
\chi(\mathcal{O}\_C(5)) = 20,
\quad 
\chi(\mathcal{O}\_{P\_{ij}}) = 1,
$$
hence $\chi(T\_C) = 5 - 20 + 10 = -5$, and since $C$ is 1-dimensional, it follows that $H^1(C,T\_C)$ is non-zero.


