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?
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 $\mathrm{Pic}(S)$.
