Let $X$ be a smooth, projective variety, $E$ a rank $r$ locally free sheaf on $X$. Fix a closed embedding $i:X \hookrightarrow \mathbb{P}^N$ and denote by $\mathcal{O}_X(m)=i^*\mathcal{O}_{\mathbb{P}^N}(m)$. For $d \gg 0$, take $r$ *general* global sections $s_1,...,s_r \in H^0(E(d))$. Denote by $Z$ the locus of points on $X$ at which the sections $s_1,...,s_r$ are linearly dependent. Denote by $D$ the singular locus of $Z$, $\tilde{X}$ the blow up of $X$ along $D$ and $\tilde{Z}$ the strict transform of $Z$ in $\tilde{X}$. Denote by $E$ the exceptional divisor in $\tilde{X}$, $F:=\tilde{Z} \cap E$ and $\pi:F \to D$ the natural morphism.

I am looking for conditions under which the morphism $\pi$ is smooth and projective. The vector bundle $E$ I am interested in, is the dual of the normal bundle $\mathcal{N}_{X|\mathbb{P}^N}$. Any idea/reference in this direction will be very helpful.