the map $\phi : \tilde{\Sigma} \rightarrow X$ has injective tangent map. And the normal sheaf $N_{\tilde{\Sigma}/X}$ is locally free along $C$ and $N_{\tilde{\Sigma}/X}\vert_C \cong {{\mathcal}{O}}(1)\oplus {{\mathcal}{O}}(1)$.

3.  The pair $(\tilde{\Sigma}, C)$ is equivalent to $({{\mathbb}{P}}^2, \text{conic})$ or $({{\mathbb}{F}}_n, \sigma)$, where ${{\mathbb}{F}}_n$ is the $n$-th Hirzebruch surface and $\sigma$ is a section of ${{\mathbb}{F}}_n \to {{\mathbb}{P}}^1$.

Note that a very free curve in $\Sigma'$ deforms to a very free curve in $X$. Indeed, it suffices to show that for two general points, the very free curve deforms to a curve that contains these two points. But one can first choose the curve $C$ to contain these two points. Then a deformation of the very free curve contains these two points.

The pair $(\tilde{\Sigma}, C)$ is not equivalent to $({{\mathbb}{P}}^2, \text{conic})$.

If the pair $(\tilde{\Sigma}, C)$ is equivalent to $({{\mathbb}{P}}^2, \text{conic})$, there exists a rational curve $L$ such that the pair $(\tilde{\Sigma}, L)$ is equivalent to the pair $({{\mathbb}{P}}^2, \text{line})$. The curve $L$ deforms to a very free curve in $X$. But in this case $-K_X \cdot L \leq 4$, which is impossible since a very free curve in a $4$-fold has $-K_X$ degree at least $5$.

If the pair $(\tilde{\Sigma}, C)$ 