Definition: A codimension r distribution $\mathcal{F}$ on a smooth complex manifold X is given by an exact sequence: $$0 \longrightarrow T_{\mathcal{F}} \longrightarrow T_{X} \longrightarrow N_{\mathcal{F}} \longrightarrow 0 $$
where $T_{\mathcal{F}}$ is a coherent sheaf of $\text{rank s}:= \text{dim(X)−r}$, and $N_{\mathcal{F}}$ is a torsion-free sheaf. The sheaves $T_{\mathcal{F}}$ and $N_{\mathcal{F}}$ are called the tangent and the normal sheaves of $\mathcal{F}$ respectively.
Consider now, $\mathcal{F}$ a codimension one distribution on $X = \mathbb{P}^{3}$ and :
$$Z = \text{Sing}(\mathcal{F}) = C \sqcup \lbrace p_{1}, \cdots, p_{n} \rbrace$$ the singular scheme of $\mathcal{F}$ with $C \subset X$ a smooth irreducible algebraic curve and closed points $p_{1}, \dots, p_{n}$.
Let $\pi : \widetilde{X} \longrightarrow X$ the blow up morphism of $X$ along of $C$ with exeptional divisor $E$.
Denote by $N_{\widetilde{\mathcal{F}}}$ the normal sheaf of the distribution $\widetilde{\mathcal{F}}$ on $\widetilde{X}$.
We know that: $$N_{\widetilde{\mathcal{F}}}^{\vee} \simeq \pi^{*}(N_{\mathcal{F}}^{\vee}) \otimes \mathcal{O}_{\widetilde{X}}(-mE) (*)$$ where $m: = \text{mult}_{E}(\pi^{*}(\mathcal{F}))$ the order of annulament of the pull-back distribution $\pi^{*}(\mathcal{F})$.
Question: With this information (*), is it possible to define the following generically surjective sheaf morphism? $$\pi^{*}(T_{X}) \stackrel{\varphi} {\longrightarrow} N_{\widetilde{\mathcal{F}}}$$
Thanks so much for any help.