Suppose we are given a scheme $S$ and two vector bundles $V$ and $W$ over $S$. Is it always true that $\mathbb{P}(V)\cong \mathbb{P}(W)$ implies that $V\cong W$ as $S$-schemes?


If the statement is false then what is the most general condition on $S$ for which it becomes true?



Are projective bundles corresponding to non-isomorphic vector bundles always non-isomorphic?
This is not true as stated, because for any line bundle $L$ on $S$ one has
$$
\mathbb{P}(V \otimes L) \cong \mathbb{P}(V),
$$
but this is the only issue. Indeed, if $X = \mathbb{P}(V) \stackrel{p}\to S$ and $S$ is connected, the relative Picard group $\mathrm{Pic}(X/S)$ is cyclic, and if $H$ is a lift to $\mathrm{Pic}(X)$ of its relatively ample generator, then
$$
p\_\*\mathcal{O}\_X(H) \cong V^\vee,
$$
and since $H$ is uniquely defined up to twist by $p^\*(\mathrm{Pic}(S))$, it follows that $V$ is uniquely defined up to twist by $\mathrm{Pic}(S)$.


