Tautological bundle on $\mathbb P^n$ is topologically trivial over the complement of a hypersurface. I'm asking this because I am confused about the first paragraph of section 1.1 of this paper: https://arxiv.org/pdf/1407.7229.pdf. In particular, I don't see why the $\mathbb C^*$-bundle $\Pi_{d,n}\setminus\Sigma_{d,n}\to N_{d,n}$ is trivial. Is this obvious? What am I missing?

The restriction of the tautological bundle ${\mathcal O}(-1)$ to the complement of a hypersurface $X \subset {\mathbb P}^n$ of degree $d > 1$ is NOT trivial. In fact, it is a generator of ${\mathrm{Pic}}({\mathbb P}^n \setminus X) \cong {\mathbb Z}/d{\mathbb Z}$.
