Computing the restriction $T_{\mathbb{P}^3|X}$ for twisted cubic in $\mathbb{P}^3$

Question

Let $i:X=\mathbb{P}^1\to\mathbb{P}^3$ be a twisted cubic given by the embedding $(u:v)\mapsto(u^3: u^2v: uv^2: v^3)$. How to compute $T_{\mathbb{P}^3|X}$?

Answer 1

It is $\mathcal{O}_{\mathbb{P}^1}(4)^3$. Starting with the exact sequence $0\to\mathcal{O}_{\mathbb{P}^3}\to\mathcal{O}_{\mathbb{P}^3}(1)^4\to T_{\mathbb{P}^3}\to 0$ and restricting, one gets, $0\to\mathcal{O}_{\mathbb{P}^1}\to\mathcal{O}_{\mathbb{P}^1}(3)^4\to\oplus_{i=1}^3\mathcal{O}_{\mathbb{P}^1}(a_i)\to 0$. So, $a_i\geq 3$ for all $i$. If $a_i=3$ for some $i$, it will force the 4 cubics to be linearly dependent and that is not true and so we get $a_i\geq 4$. But $12=\sum a_i$ implies all the $a_i$s must be 4.