Normal bundle to complete intersection in $\mathbb{P}^n$ 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 $\bigoplus\limits_{i=1}^k\mathcal{O}_X(d_i)$?
 A: 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 
$$
0\to\mathcal{T}_X\to\mathcal{T}_{\mathbb{P}^n}|_X\to N\to0
$$
A: By definition, the normal bundle is $(\mathscr I/\mathscr I^2)^\vee$, where $\mathscr I$ is the ideal sheaf of $X$.
Since $X$ is a complete intersection, the ideal sheaf $\mathscr I$ is resolved by the Koszul complex. I.e., we have an exact sequence of the form
$$
0 \to \mathscr K \to \bigoplus_{i,j} \mathscr O(-d_i-d_j) \xrightarrow{\varphi} \bigoplus \mathscr O(-d_i) \to \mathscr I \to 0.
$$
Where $\varphi$ is multiplication by the vector $(f_1,\ldots, f_n)$. We have that $\mathscr I / \mathscr I^2 \simeq \mathscr I \otimes_{\mathscr O_{\mathbb P^n}} \mathscr O_X$. Thus, tensoring the above sequence with $\mathscr O_X$ we get
$$
\mathscr K \otimes \mathscr O_X \to \bigoplus_{i,j} \mathscr O_X(-d_i-d_j) \xrightarrow{\varphi} \bigoplus \mathscr O_X(-d_i) \to \mathscr I/\mathscr I^2 \to 0.
$$
But the middle map $\varphi$ is zero in $\mathscr O_X$. Hence
$$
\bigoplus \mathscr O_X(-d_i) \simeq \mathscr I/\mathscr I^2.
$$
Dualizing, we get the result.
