Definition 1): A codimension one distribution of degree $d \geq 0$ in $\mathbb{P}^{n}$ is given by an exact sequence: $$\mathscr{F}: 0 \longrightarrow T_{\mathscr{F}} \longrightarrow T\mathbb{P}^{n}\longrightarrow \mathcal{I}_{Z/ \mathbb{P}^{n}}(d+2) \longrightarrow 0$$ where :
I) $T_{\mathscr{F}}$ is coherent sheaf of rank $s = n - 1$. (tangent sheaf of $\mathscr{F}$)
II) $\mathcal{I}_{Z/ \mathbb{P}^{n}}$ is the ideal sheaf of singular scheme $Z \subset \mathbb{P}^{n}$ of $\mathscr{F}$ with $\text{codim}(\text{Sing}(Z)) \geq 2$.
III) $N_{\mathscr{F}} = \mathcal{I}_{Z/ \mathbb{P}^{n}}(d+2)$ is a torsion free sheaf. (Normal sheaf of $\mathscr{F}$)
Definition 2): A codimension one foliation is an integrable distribution ($\omega \wedge d\omega = 0$) where $\omega \in H^{0}(\mathbb{P}^{n}, \Omega_{\mathbb{P}^{n}}^{1}(d+2))$ induces $\mathscr{F}$.
Consider $n = 2$ and $\pi : \widetilde{\mathbb{P}^{2}} \longrightarrow \mathbb{P}^{2}$ the blowup morphism in $p \in \mathbb{P}^{2}$ and $E = \pi^{-1}(p)$ the exceptional divisor.
Let $\mathscr{F}$ the foliation in $\mathbb{P}^{2}$ induces by $\omega$ (because, in this case, we have: $\omega \wedge d\omega = 0$. Frobenius Integrability Condition). I was able to show the following isomorphisms:
1) $N_{\pi^{*}\mathscr{F}}^{*} \simeq \pi^{*}(N_{\mathscr{F}}^{*}) \otimes \mathcal{O}_{\widetilde{\mathbb{P}^{2}}}(\ell E)$ (conormal sheaf of $\widetilde{\mathscr{F}} = \pi^{*}\mathscr{F}$)
2) $T_{\pi^{*}\mathscr{F}} \simeq \pi^{*}(T_{\mathscr{F}}) \otimes \mathcal{O}_{\widetilde{\mathbb{P}^{2}}}((\ell - 1)E)$ (tangent sheaf of $\widetilde{\mathscr{F}} = \pi^{*}\mathscr{F}$)
where $\ell : = \text{mult}_{E}(\pi^{*}\mathscr{F})$ the order of annulment of pulback foliation $\pi^{*}\mathscr{F}$ at $E$.
First question : In the literature, the degree of a coherent sheaf is defined as: $\text{deg}(\mathcal{F}): = \text{deg}(\text{det}(\mathcal{F})) $. In this case, what would it be $\text{deg}(N_{\pi^{*}\mathscr{F}}^{*})$ and $\text{deg}(T_{\pi^{*}\mathscr{F}})$?
For $n = 3$. Let $\pi: \widetilde{\mathbb{P}^{3}} \longrightarrow \mathbb{P}^{3}$ be the blowup of $\mathbb{P}^{3}$ along a regular curve $C$ with exceptional divisor $E$. (Here we have a distribution, because not necessarily, we have: $\omega \wedge d\omega = 0$).
In this case, it was not possible to show isomorphisms analogous to items (1) and (2) for $T_{\pi^{*}\mathscr{F}}$ and $N_{\pi^{*}\mathscr{F}}^{*}$.
Second question Does the degree of the $ C $ curve come into play to determine $N_{\pi^{*}\mathscr{F}}^{*}$ and $T_{\pi^{*}\mathscr{F}}$ in this case? And the degree of these sheaves, how is it calculated in this case?
Thanks in advance and any suggestions, references will be most welcome.