Is the tensor product a reflexive sheaf? A dimension one distrbution $\mathscr{F}$ on $X =\mathbb{P}^{3}$ of degree $d \geq 0$ is given by an exact sequence : $$\mathscr{F} : 0 \longrightarrow \mathcal{O}_{\mathbb{P}^{3}}(1-d) \longrightarrow T\mathbb{P}^{3} \longrightarrow I_{Z} \otimes N_{\mathscr{F}}  \longrightarrow 0$$ where $I_{Z}$ is the ideal sheaf of a subscheme $Z \subset \mathbb{P}^{3}$, the singular scheme of $\mathscr{F}$ and $N_{\mathscr{F}}$ is called of normal sheaf of $\mathscr{F}$. By $\mbox{R. Hartshorne}$, (Stable Reflexive Sheaves, corollary 1.2), $(I_{Z} \otimes N_{\mathscr{F}})^{*}$ is a reflexive sheaf. 
Dualizing the sequence above, we have: 
\begin{equation}
0 \longrightarrow (I_{Z} \otimes N_{\mathscr{F}})^{*} \longrightarrow \Omega_{X}^{1} \longrightarrow \mathcal{O}_{\mathbb{P}^{3}}(d-1) \longrightarrow \mbox{ext}^{1}(I_{Z} \otimes N_{\mathscr{F}}, \mathcal{O}_{\mathbb{P}^{3}}) \longrightarrow \cdots
\end{equation}
1) $\mbox{Codim}(Z) \geq 2$ (this information is given)
2) Under what hypothesis the sheaf $I_{Z} \otimes N_{\mathscr{F}}$ is reflexive?
3) For a coherent sheaf $F$ over a $n$-dimensional complex manifold $X$, we have :$\mbox{dim}\left({supp}(ext^{i}(F, \mathcal{O}_{X})\right) \leq n - i$. (Lemma 1.1.2, Christian Okonek)
4) $\mbox{dim}\left({supp}(ext^{1}(I_{Z} \otimes N_{\mathscr{F}} , \mathcal{O}_{X})\right)$?
Can someone help me?
Thank you in advance.
 A: I will make this answer geometrically towards distributions. 
First we have the exact sequence $$ 0 \rightarrow \mathcal{F} \rightarrow TX \rightarrow N_{\mathcal{F}} \rightarrow 0$$ defining a distribution. Here $N_{\mathcal{F}} = TX/\mathcal{F}$. 


*

*A torsion local section of $N_{\mathcal{F}}$ is a germ of vector field $v$ such that $v$ is not tangent to the distribution, but there exists some function germ $f$ such that $fv\in \mathcal{F}$. Then the distribution is singular along $\{f=0\}$. Since this set is of codimension one, we can avoid this situation trivially reducing to distributions with singularities only in codimension $\geq 2$. 

*We say that a foliation is reduced (in the sense of Suwa) if for every open subset $U\subset X$ we have
$$
\Gamma(U, TX) \cap \Gamma(U\backslash S(\mathcal{F}), \mathcal{F}) = \Gamma(U, \mathcal{F})
$$
where $S(\mathcal{F})$ is the singular set. This means precisely that a vector field tangent to the distribution away from the singular set must come from a section of the subsheaf $\mathcal{F}$. This amounts to $N_{\mathcal{F}}$ being normal in the sense of Barth (second isomorphism theorem for modules).


By the Proposition 1.6 of Hartshorne's paper Stable Reflexive Sheaves, a coherent sheaf  is reflexive if and only if it is torsion free and normal in the sense of Barth. 
Therefore the distribution is reduced if and only if $N_{\mathcal{F}}$ is reflexive.
Also, see Suwa's paper for these questions concerning the $Ext$.
