Computing canonical sheaf

Question

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?

Thank you in advance.

Answer 1

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.