The exercise is quoted below and here is what I have done so far.
First off, (a) is just a definition there is nothing to show. For (b), the exercise is a straightforward check of the definitions using the construction of the projective space bundle i.e. use the definition of $\mathbb{P}(\mathscr{E})$ and verify the assertions in (a). Part (d) follows easily from part (c) and (b) and combined with Hartshorne Exercise II.7.9 which states $\mathbb{P}(\mathscr{E})$ is not changed if $\mathscr{E}$ is replaced with $\mathscr{E}\otimes\mathscr{L}$ where $\mathscr{L}$ is invertible.
So the heart of the exercise is in (c). A hint is provided by Hartshorne and it is not exactly trivial. First off, $\mathscr{L}_0=\mathcal{O}(1)$ on $U\times\mathbb{P}^n$ is doable since $X$ is regular and we know the Picard group is $\operatorname{Pic}(U)\times\mathbb{Z}$ in this case by the last section (this is where I use regular schemes are normal schemes) and we have used regularity in a nontrivial manner here.
Explanation of where I am: Hartshorne suggests we extend $\mathscr{L}_0$ to an invertible sheaf on $X$. There is no reason to expect that I can use the transition functions with $\mathscr{L}_0$ is extend the sheaf. So I suspect that I choose $\mathcal{O}(1)$ on finitely many $U_1,\dots,U_\ell$ and show that they glue to an invertible sheaf on $P$ in this way. The transition functions are $A$-linear automorphisms so they should preserve $\mathcal{O}(1)$ over each intersection $U_i\cap U_j$. This feels a bit hazy to me.
What I am stuck with: Is the previous argument in the right direction? Afterwards, I do not see how to show $\pi_*\mathscr{L}$ is a locally free sheaf nor do I see a path towards proving $P=\mathbb{P}(\pi_*\mathscr{L})$.
I also suspect "regular" is needed, but until I solve the first part, I am not confident in that suspicion.
Hints only please.
Hartshorne II.7.10 $\mathbb{P}^{n}$-Bundles Over a Scheme. Let $X$ be a noetherian scheme.
(a) By analogy with the definition of a vector bundle (Ex. 5.18), define the notion of a projective $n$-space bundle over $X$, as a scheme $P$ with a morphism $\pi: P \rightarrow X$ such that $P$ is locally isomorphic to $U \times \mathbb{P}^{n}, U \subseteq X$ open, and the transition automorphisms on $\operatorname{Spec} A \times \mathbb{P}^{n}$ are given by $A$-linear automorphisms of the homogeneous coordinate ring $A\left[x_{0}, \ldots, x_{n}\right]$ (e.g., $x_{i}^{\prime}=\sum a_{i j} x_{j}, a_{i j} \in A$).
(b) If $\mathscr{E}$ is a locally free sheaf of rank $n+1$ on $X$, then $\mathbb{P}(\mathscr{E})$ is a $\mathbb{P}^{n}$-bundle over $X .$
(c) Assume that $X$ is regular, and show that every $\mathbb{P}^{n}$-bundle $P$ over $X$ is isomorphic to $\mathbb{P}(\mathscr{E})$ for some locally free sheaf $\mathscr{E}$ on $X$.
[Hint: Let $U \subseteq X$ be an open set such that $\pi^{-1}(U) \cong U \times \mathbb{P}^{n}$, and let $\mathscr{L}_{0}$ be the invertible sheaf $\mathcal{O}
(1)$ on $U \times \mathbb{P}^{n}$. Show that $\mathscr{L}_{0}$ extends to an invertible sheaf $\mathscr{L}$ on $P$. Then show that $\pi_{*}\mathscr{L}=\mathscr{E}$ is a locally free sheaf on $X$ and that $\left.P \cong \mathbb{P}(\mathscr{E}) .\right]$Can you weaken the hypothesis "$X$ regular"?
(d) Conclude (in the case $X$ regular) that we have a 1-1 correspondence between $\mathbb{P
}^{n}$-bundles over $X$, and equivalence classes of locally free sheaves $\mathscr{E}$ of rank $n+1$ under the equivalence relation $\mathscr{E}' \sim \mathscr{E}$ if and only if $\mathscr{E}' \cong \mathscr{E} \otimes \mathscr{M}$ for some invertible sheaf, $\mathscr{M}$ on $X$.
Best Answer
You're kind of on the right track here, but you're definitely a little stuck and I agree that this problem can be a little opaque at first (especially in regards to how to actually extend $\mathcal{L}_0$).
One of the biggest advantages to working in a situation where there's a hypothesis like regularity around is the fact that you can swap between Weil divisors, Cartier divisors, and line bundles depending on which one is easier to think about or work with. For instance, $\mathcal{L}_0=\mathcal{O}_{\Bbb P^n_U}(1)$ on $\Bbb P^n\times U$ corresponds to a hyperplane $H_0\subset \Bbb P^n\times U$. What happens if you consider the closure $H$ of $H_0$ in $X$ and look at the line bundle $\mathcal{L}$ on $X$ corresponding to $H$? You should be able to make some good progress here via considering the restriction of this to $U$ and sets like it (there will be some work involved, but you requested hints and so I'm not going to tell you everything).
For the second part about showing $\pi_*\mathcal{L}$ is locally free, refer back to exercise II.7.9 for some valuable information about Picard groups of things like $\Bbb P(\mathcal{E})$. From here you should know some sort of formula involving pushforwards and pullbacks from an exercise earlier in chapter II which makes talking about the pushforward of $\mathcal{L}$ easier. Next, to verify $\Bbb P(\pi_*\mathcal{L})$ is the right thing, try to construct a global map that you can check is an isomorphism locally.
Finally, if you're interested in whether/how much regularity can be weakened, you may be interested to check out this MSE answer from a year and change ago.