# canonical divisor from canonical sheaf

Let $$X$$ be a complete intersection of $$m$$ hypersurfaces in $$P^n$$ over some field $$k$$. I have computed that the canonical sheaf is $$\omega_X=\mathcal{O}_X(\sum d_i-n-1)$$, where $$d_i$$ are the degrees of the hypersurfaces. I also know that the canonical divisor of a hypersurface $$Y$$ is $$K_Y=(d-n-1)H$$ if $$Y$$ is a hypersurface of degree $$d$$.

How should I compute the canonical divisor of a complete intersection of hypersurfaces? It is probably something simple that I didn't observe. Thanks in advance.

• better not say some field $k$, perhaps $char(k)$ dividing $d$ could cause some problems. Commented May 2, 2023 at 7:11

Let $$H$$ be a hyperplane section of $$Y$$, such that the embedding $$\iota$$ of $$Y$$ into $$\mathbb{P}^n$$ is given by $$|H|$$. Then by definition $$\mathcal{O}_X(1)=\iota^*\mathcal{O}_{\mathbb{P}^n}(1)=\mathcal{O}_X(H)$$. Hence $$\omega_X=\mathcal{O}_X(-n-1+\sum{d_i})=\mathcal{O}_X((-n-1+\sum{d_i})H)$$. Therefore, the canonical divisor (which is only determined up to linear equivalence) is given by $$K=(-n-1+\sum{d_i})H$$.