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It is well known that we have an isomorphism $\mathrm{Pic}(\mathbb{P^{1}} \times \mathbb{P^{1}}) \cong \mathbb{Z} \oplus \mathbb{Z}$. Does anyone know how to determine the type of the canonical divisor $\omega_{\mathbb{P^{1}} \times \mathbb{P^{1}}}$, that is, its image in $\mathbb{Z} \oplus \mathbb{Z}$?

I am studying Hartshorne's book Algebraic Geometry, where this question appears in the Example II.$8.20.3$ on page $183$, and the isomorphim is that in the Example II.$6.6.1$ on page $135$. Hartshorne states that the type os the canonical divisor is $(-2, -2)$, but how to prove? It seems difficult for me.

Thanks.

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The canonical line bundle on the product is the exterior tensor product of the canonical line bundles on each factor. This should reduce you to computing the canonical bundle on $\mathbb P^1$, which I would guess you know.

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