Canonical bundle of a regular projective complete intersection ($K_V\cong \mathcal{O}({\sum }_i{d}_i-n-1){|}_V$)

For regular homogeneous equations of degrees $d_1,\dots ,d_r$ in ${\mathbb{P}}^n$, the normal bundle of a regular zero locus is ${⨁}_i\mathcal{O}(d_i){|}_V$. 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.