I encountered what looks like a contradiction to the Birkhoff-Grothendieck theorem, so I am hoping somebody can point out where I am confused.
Let $\ell \subset V \subset \mathbb{P}^n$ be a line contained in a hypersurface of degree $d$. In $\mathbb{P}^n$, $\ell$ is the transverse intersection of $n-1$ hyperplanes, so its normal bundle is $$N_{\mathbb{P}^n/\ell} \cong (n-1)[H]\big|_\ell = \mathcal{O}(1)^{\oplus n-1}.$$ But we also have $$N_{\mathbb{P}^n/\ell} \cong N_{V/\ell} \oplus N_{\mathbb{P}^n/V} = N_{V/\ell} \oplus d[H]\big|_\ell = N_{V/\ell} \oplus \mathcal{O}(d).$$ By Birkhoff-Grothendieck, $N_{V/\ell}$ factors as a sum of line bundles, giving a factorization of $N_{\mathbb{P}^n/\ell}$, and further that factorization is supposed to be unique up to reordering. How does one factorization have an $\mathcal{O}(d)$ summand while the other does not?
Let me explain the error in the arguments above. The normal bundle to $\ell$ is not a direct sum$$N_{\ell/V} \oplus N_{V/\mathbb{P}^n}.$$For this to be true the line should be a section of $V$ by a some linear $2$-space, which is possible only if$$\deg V = 1.$$