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 or has only nodes. For context, a positive answer would imply that $C$ has no locally trivial deformations.

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.