TITLE: Normal bundle to complete intersection in $\mathbb{P}^n$
QUESTION [10 upvotes]: Let $X\subset\mathbb{P}^n$ be a complete intersection defined by irreducible polynomials $f_1,...,f_k$ of degrees $d_1,...,d_k$. How to show that the normal bundle of $X$ is isomorphic to $\bigoplus\limits_{i=1}^k\mathcal{O}_X(d_i)$?

REPLY [2 votes]: Since every $f_i$ is a section in $H^{0}(\mathbb{P}^n,\mathcal{O}(d_i))$, denote $F=\oplus{f_i}$ the section in $\oplus\mathcal{O}(d_i)$, obviously $Z(F)=X$. Now restric $dF$ to $\mathcal{T}_{\mathbb{P}^n}|_X$, we get an exact  sequence:
$$
0\to\mathcal{T}_X\to\mathcal{T}_{\mathbb{P}^n}|_X\to\oplus\mathcal{O}_X(d_i)\to0
$$
Compare with 
$$
0\to\mathcal{T}_X\to\mathcal{T}_{\mathbb{P}^n}|_X\to N\to0
$$