# Computing canonical sheaf

I am learning about canonical sheaf. I am struggling with computing the canonical sheaf of:

1. Del pezzo surface $$X$$: I know that such surface is either $$\mathbb{P}^1 \times \mathbb{P}^1$$ or the blow-up of $$\mathbb{P}^2$$ at most $$8$$ points. But I do not know how to compute the canonical sheaf of $$X$$.
2. I know that if $$X=H_1\cap ...\cap H_r$$ is the complete intersection of codimension $$r$$ in $$\mathbb{P}^n$$ with $$\deg\, H_i =d_i$$ then the canonical sheaf of $$X$$ is $$\mathcal{O}_X(d_1+\cdots+d_r -n -1)$$. Why this is true?

For the blow up of $$\mathbb{P}^2$$ iin $$k$$ points t is $$-3 \pi^*H + E_1 \ldots + E_{k}$$ ($$H$$ is the hyperplane class, $$\pi$$ is the blow down map to $$\mathbb{P}^2$$, $$E_k$$ are the exceptional divisors).

This follows from a general fact that the canonical divisor of $$KBl_{p}(S) = \pi^*{K_S} + E$$ where is the exceptional divisor and the on $$\mathbb{P}^2$$ the canonical sheaf is $$-3H$$ (which in turn follows from the Euler sequence on projective space).

For $$\mathbb{P}^1 \times \mathbb{P}^1$$ it is $$-2 (\mathbb{P}^1 \times \{p_0\} + \{p_0\} \times \mathbb{P}^1)$$.

For complete intersections the formula follows from the adjunction formula (applied $$r$$ times each time getting a formula for the intersecion with a hypersurface).

In all of the descriptions above I am implicitly using the correspondane between divisors and locally free sheaves.

All these facts can be found in an introductory graduate text in algebraic geometry. I recommend "complex algebraic surfaces" by Beauville.

• Thank you for your answer and a good reference. I write down the case of complete intersection as following: Since $H_1$ is in codimension 1 then by adjunction formula we have $\omega_{H_1}=\omega_{\mathbb{P}^n} \otimes \mathcal{O}(H_1)\otimes \mathcal{O}_{H_1}=\mathcal{O}_{H_1}(d_1-n-1)$. Now apply the formula again for $H_1\cap H_2$ this give $\omega_{H_1\cap H_2}=\omega_{H_1} \otimes \mathcal{O}_{H_1}(H_1.H_2)\otimes \mathcal{O}_{H_1 \cap H_2}.$ Can you explain for me why $\mathcal{O}_{H_1}(H_1.H_2) \cong \mathcal{O}_{H_1}(d_2)$ ?? Do you have any suggest for me to the case $P^1\times P^1$
– Hong
Commented Apr 7, 2022 at 16:58