Let $X = \mathbb{P}^{3}$ and $C \subset X$ an irreducible smooth curve. Let $I_{C}$ be the ideal sheaf of $C$.
I am trying to calculate the following cohomology: $$H^{i}(X, \Omega_{X}^{1} \otimes \bigwedge^{i+1}T_{X} \otimes S^{i}(\mathcal{F}) \otimes I_{C}(m))$$ with $i = 1, 2$ and $0 < m \leq 2$, where:
1) $\mathcal{F}$ is a locally free sheaf of $\text{rank}$ one. More precisely, in my case, it is: $\mathcal{F} = \mathcal{O}_{X}(c + d)$ and $S^{i}(\mathcal{F})$ is the $\text{i-th}$ symmetric power of $\mathcal{F}$.
2) $I_{C}$ is the ideal sheaf of $C$.
My objective is: $H^{i}(X, \Omega_{X}^{1} \otimes \bigwedge^{i+1}T_{X} \otimes S^{i}(\mathcal{F}) \otimes I_{C}(m)) = 0$ with $i = 1, 2$
What i'm trying to do: For a question asked here in the forum: (https://math.stackexchange.com/users/715318/emanuell), Holomorphic Vector Bundle and Non-Degenerate Pairing, I have the following isomorphisms:
a) For $i = 1$. $$\bigwedge^{2}T_{X} \simeq \Omega_{X}^{1}(4)$$
b) For $i = 2$ $$\bigwedge^{3}T_{X} \simeq \mathcal{O}_{X}(4)$$
My problem now is the $S^{i}(\mathcal{F})$. I don't know what is: $\bigwedge^{i+1}T_{X} \otimes S^{i}(\mathcal{F})$ (*)
After understanding this (*), I will try to use the following tools: Mumford's Regularity Theorem, because:
i) Bott's formula tells us that $\Omega_{X}^{1}$ is 2-regular .
ii) The corollary 2.1 of the article: Vanishing Theorems, A Theorem Of Severi and the equations defining projective variety, that says about the aforementioned regularity for an ideal sheaf.
Thank you and a lot from now, any help.