# ag.algebraic geometry – Hypersurfaces in projective bundles over \$mathbb P^1\$

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 $$mathbb P(E)$$?