Let $C\subset\mathbb{P}^2\subset\mathbb{P}^n$ be a smooth conic (everything is over the field $\mathbb{C}$). I want to compute $T_{\mathbb{P}^n|C}$ and $N_{C/\mathbb{P}^n}$.
Let $z_0,z_1,...,z_n$ be a homogeneous coordinates in $\mathbb{P}^n$ and suppose that $\mathbb{P}^2$ is given by $\{z_3=...=z_n=0\}$ and $C=\{z_0^2+z_1^2+z_2^2=0\}$. To compute $T_{\mathbb{P}^n|C}$ I use Euler sequence
$$0\to\mathcal{O}_{\mathbb{P}^n}\to\mathcal{O}_{\mathbb{P}^n}(1)^{\oplus(n+1)}\to T_{\mathbb{P}^n}\to0.$$ Tensoring it with $\mathcal{O}_C$ we obtain the exact sequence $$0\to\mathcal{O}_{C}\stackrel{\left( \begin{array}{c} z_0\\ z_1\\ z_2\\ 0\\ \vdots\\ 0\\ \end{array} \right)}\to\mathcal{O}_{C}(1)^{\oplus(n+1)}\to T_{\mathbb{P}^n|C}\to0,$$ from which we have that $T_{\mathbb{P}^n|C}=F\oplus\mathcal{O}_{C}(1)^{\oplus(n-2)}$, where $F$ is the cokernel of the map $$\mathcal{O}_{C}\stackrel{\left( \begin{array}{c} z_0\\ z_1\\ z_2 \end{array} \right)}\to\mathcal{O}_{C}(1)^{{\oplus3}}.$$ Then $F$ is equal to $\mathcal{O}_{C}(1)\oplus\mathcal{O}_{C}(2)$ or $\mathcal{O}_{C}\oplus\mathcal{O}_{C}(3)$. What is the correct answer?
To compute $N_{C/\mathbb{P}^n}$ I use the exact sequence $$0\to T_C\to T_{\mathbb{P}^n|C}\to N_{C/\mathbb{P}^n}\to0.$$ But $T_C=\mathcal{O}_C(1)$ so it seems that $N_{C/\mathbb{P}^n}=F\oplus\mathcal{O}_{C}(1)^{\oplus(n-3)}$. Is it correct?