= Canonical bundle of a regular projective complete intersection
{title2=$K_V\cong\mathcal O(\sum_i d_i-n-1)|_V$}
For regular homogeneous equations of degrees $d_1,\ldots,d_r$ in $\mathbb P^n$, the <normal bundle of a regular zero locus> is $\bigoplus_i\mathcal O(d_i)|_V$. The <Euler sequence> gives $K_{\mathbb P^n}=\mathcal O(-n-1)$, hence $K_V=\mathcal O(\sum_i d_i-n-1)|_V$. Equations with unreduced multiplicities do not have this conclusion for the underlying reduced submanifold.
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