Definition: Let $X$ be a smooth complex projective variety of dimension $n$. A holomorphic distribution of rank $k$ on $X$ is nonzero coherent subsheaf $\mathcal{F} \subsetneq T_{X}$ of generic rank $k$ which is saturated, i.e, such that $T_{X}/ \mathcal{F}$ is torsion free.
We can also define a distribution above by the following exact sequence: $$0 \longrightarrow T_{\mathcal{F}} \longrightarrow T_{X} \longrightarrow N_{\mathcal{F}} \longrightarrow 0 \tag{$1$}$$ Denote by $Z = \text{Sing}(\mathcal{F})$ the singular scheme of $\mathcal{F}$.
Let $\pi : \widetilde{X} \longrightarrow X$ be the blow up morphism of $X$ along of an irreducible smooth curve $C \subset X$ with exceptional divisor $E$.
Question: It is possible to define a morphism : $\pi^{*}(T_{X})\stackrel{f} \longrightarrow N_{\widetilde{\mathcal{F}}}$ surjective?
where: $N_{\widetilde{\mathcal{F}}}$ is the normal sheaf of $\widetilde{\mathcal{F}}$ the distribution in $\widetilde{X}$ induced by $\pi$.
I don't know if it makes sense but what I tried to do I write below.
What I'm trying to do: In the complement of $Z$ we have the exact sequence ($1$) is an exact sequence between locally free sheaves. So, since pullback is an exact functor when applied to locally free sheaves, we have: $$0 \longrightarrow \pi^{*}(T_{\mathcal{F}}) \longrightarrow \pi^{*}(T_{X}) \longrightarrow \pi^{*}(N_{\mathcal{F}}) \longrightarrow 0 \tag{$2$}$$
It can be shown that: $N_{\widetilde{\mathcal{F}}} \simeq \pi^{*}(N_{\mathcal{F}}) \otimes \mathcal{O}_{\widetilde{X}}(-\ell E) \tag{$3$}$
where $\ell = \text{mult}_{E}(\pi^{*}\mathcal{F})$ the order of annulment of the pullback distribution $\pi^{*}(\mathcal{F})$ at $E$.
Twisting the exact sequence $(2)$ by $\mathcal{O}_{\widetilde{X}}(-\ell E)$ (writing the part that interests us) we have $$\pi^{*}(T_{X}) \otimes \mathcal{O}_{\widetilde{X}}(-\ell E) \stackrel{\varphi} \longrightarrow \pi^{*}(N_{\mathcal{F}})\otimes \mathcal{O}_{\widetilde{X}}(-\ell E) \rightarrow 0 \tag{$4$}$$
By $(3)$ we have the surjective morphism of locally free sheaves: $$\pi^{*}(T_{X}) \otimes \mathcal{O}_{\widetilde{X}}(-\ell E) \longrightarrow N_{\widetilde{\mathcal{F}}} \longrightarrow 0 \tag{$5$}$$
The most I have so far is $(5)$.
Second question: Would it be absurd to hypothesize the existence of morphism $f$? Otherwise, how to explain the existence of such a morphism satisfying the desired property?
Any suggestions, references, help, will be welcome.
Thank you very much.