Suppose $X$ is a scheme (possibly smooth / $\mathbb C$), $E$ is a vector bundle of rank $e$ on $X$ with corresponding projective bundle $p:\mathbb PE \to X$, and $s: X \to \mathbb PE$ is a section. I wanted to compute the normal bundle $N$ of $s(X) \subset \mathbb PE$ in terms of the short exact sequence $$\tag{1}\label{eq1}0\to K \to E \to L \to 0$$ which corresponds to $s$. I know that the correct result is $$N \cong K^\vee \otimes L,$$ but somehow my computation only yields $K^\vee$, and I don't know what went wrong. Here is what I did:
Taking symmetric powers of \eqref{eq1} yields $$\label{eq2}\tag{2} 0 \to I \to S^* E \to S^*L \to 0,$$ together with a graded¹ surjection $\varphi: K \otimes S^*E[-1] \to I$. We want to compute $N = (I / I^2)^\vee$. A local computation² shows that $I^2$ is generated by the image of $(K \otimes S^*E[-1])_{\geq 2}$, and that $K = (K \otimes S^* E[-1])_1$ is mapped isomorphically to $I_1$. Hence $$I / I^2 \cong K \otimes S^* E[-1] / (K \otimes S^* E[-1])_{\geq 2} \cong K.$$ At which step do I have to take $L$ into account?
¹ Convention: $(M[k])_d = M_{d+k}$.
² If one takes an open affine $\operatorname{Spec} A \subset X$ over which $K, E$, and $L$ xivare free, \eqref{eq1} becomes $$0 \to A^{e-1} \to A^e \xrightarrow{p_e} A \to 0,$$ where the first map is $(a_1, \dotsc, a_{e-1}) \mapsto (a_1, \dotsc, a_{e-1}, 0)$, and the second map is projection to the last component. In \eqref{eq2} we then have polynomial rings $$0 \to (t_1, \dotsc, t_{e-1}) \to A[t_1, \dotsc, t_e] \to A[t_e] \to 0.$$ The map $\varphi$ is $$\bigoplus_{i=1}^{e-1} A[t_1, \dotsc, t_e] \cdot e_i \to I, \quad e_i \mapsto t_i.$$