I am working on a suggestion of a comment here.

Let $E rightarrow mathbb P^1$ be a *non-trivial* vector bundle of rank $r$ with $deg E =0$ and $mathbb P(E) rightarrow mathbb P^1$ be its projectization.

The question is:

Is there a section of $mathcal O_{mathbb P(E)} (r)$ that admits a crepant resolution for some $E$?

For example, let $E = mathcal O oplus cdots mathcal O oplus mathcal O(-1) oplus mathcal O(1)$,

Is there a section of $mathcal O_{mathbb P(E)} (r)$ that admits a crepant resolution for some $r$?

What if $r=4$ or $5$?

If $E$ is trivial, then $mathbb P(E) = mathbb P^{r-1} times mathbb P^1$ and its hypersurface can be express by bihomogeneous polynomials.

In case that $E$ is not trivial, is there some similar concrete way to describe hypersurfaces in $mathbb P(E)$?