hirzebruchSurface a
The $a$-{th} Hirzebruch surface is a smooth projective normal toric variety. It can be defined as the $\PP^1$-bundle over $X = \PP^1$ associated to the sheaf ${\mathcal O}_X(0) \oplus {\mathcal O}_X(a)$. It is also the quotient of affine $4$-space by a rank two torus.
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When $a = 0$, we obtain $\PP^1 \times \PP^1$.
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The map from the group of torus-invariant Weil divisors to the class group is chosen so that the positive orthant corresponds to the cone of nef line bundles.
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