Restriction of relative sheaf on a section suppose $X\rightarrow Y$ is a projective bundle or rank $r$.
Let $S$ be a section of this bundle.
What is the restriction of the relative tangent bundle $T_{X/Y}$ on the section $S$ ? Is it free ?
 A: It doesn't matter whether $Y$ is smooth actually... Let's say it's noetherian for safety. 
Let $\pi: X=\mathbb P(\mathscr E)\to Y$ where $\mathscr E$ is a rank $r+1$ locally free sheaf on $Y$ and let $\mathscr O_{\mathbb P(\mathscr E)}(1)$ be the associated relative very ample line bundle. Then one has the natural short exact sequence:
$$
0 \to \Omega_{X/Y} \to \pi^*\mathscr E\otimes \mathscr O_{\mathbb P(\mathscr E)}(-1) \to \mathscr O_{\mathbb P(\mathscr E)} \to 0
$$
Now let $\sigma: Y\to X$ be a section. Recall that sections of projective bundles are in one-to-one correspondence with surjections to line bundles: $\mathscr E\to \mathscr L$. Restricting the above short exact sequence to the section is the same as pulling it back to $Y$ via $\sigma$, so we get that the restricted sequence is
$$
0 \to \Omega_{X/Y}|_{\sigma(Y)} \to \mathscr E\otimes \sigma^*\mathscr O_{\mathbb P(\mathscr E)}(-1) \to \mathscr O_{Y} \to 0
$$
By the above correspondence between sections and surjective maps $\sigma^*\mathscr O_{\mathbb P(\mathscr E)}(1)\simeq \mathscr L$ and the map on the right hand side of the above sequence corresponds to the surjective map $\mathscr E\to \mathscr L$.
So we get that we may interpret the above sequence the following way:
$$
0 \to \Omega_{X/Y}|_{\sigma(Y)} \to \mathscr E\otimes \mathscr L^{-1} \to \mathscr O_{Y} \to 0
$$
so the restriction $T_{X/Y}|_{\sigma(Y)}$ is the cokernel of the map 
$\mathscr O_Y\to \mathscr Hom_Y(\mathscr E, \mathscr L)$ that maps $1\in \mathscr O_Y$ to the surjective map corresponding to the section $\sigma$. This may or may not be free depending on the bundle and the section. 
