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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.


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up vote 4 down vote accepted

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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