Normal bundle to complete intersection in $\mathbb{P}^n$

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.

Tags:

Algebraic Geometry

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