Is the tensor product a reflexive sheaf?

Question

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.

Answer 1

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 $N_{\mathcal{F}} = TX/\mathcal{F}$.

1. 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 $\geq 2$.
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 $Ext$.