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There are several ways of describing a Hirzebruch surface, for example as the blow-up of $\mathbb{P}^2$ at one point or as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$.

I am wondering if these surfaces admit a three-dimensional analogue and in particular if the blow-up of $P^3$ at one point is the same thing as $\mathbb{P}(\mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1} \oplus \mathcal{O}_{\mathbb{P}^1}(n))$ (if not, is this projectivization even toric ?). I have a lacunar background in algebraic geometry and I don't have any idea on how to attack the problem. Do you have any suggestions ?

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    $\begingroup$ $\mathrm{Bl}_{p}(\mathbb{P}^3) \cong \mathbb{P}_{\mathbb{P}^2}(\mathcal{O} \oplus \mathcal{O}(-1))$. $\endgroup$
    – Sasha
    Nov 28, 2023 at 10:14

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The 3-dimensional analogues should be $\mathbb{P}(O(a)\oplus O(b)\oplus O(c))$, yes. These are exactly the $\mathbb{P}^2$-bundles over $\mathbb{P}^1$.

In general, any projectivization of a vector bundle on $\mathbb{P}^1$ is toric. (See the book by Cox, Little, Schenck). If I remember correctly, all smooth toric varieties of picard number 2 is of this form.

The blow-up of $\mathbb{P}^3$ at a point is however not of the $\mathbb{P}(O(a)\oplus O(b) \oplus O(c))$, because the blow-up does not even admit a morphism to $\mathbb{P}^1$.

However, it is a projective bundle over $\mathbb P^2$; projection from a point gives a morphism $$Bl_p\mathbb{P}^3\to \mathbb P^2$$ which is a $\mathbb {P}^1$-bundle over $\mathbb P^2$. Explicitly, it is given by $\pi:\mathbb{P}(O \oplus O(1))\to \mathbb P^2$.

Finally, $\mathbb{P}(O \oplus O \oplus O(1))$ defines the blow-up of $\mathbb{P}^3$ along a line. (Using the Hartshorne notation for $\mathbb P(\mathcal E)$).

The proofs of these statements are similar to the surface case.

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  • $\begingroup$ Thank you for this very nice answer. $\endgroup$
    – Yromed
    Nov 29, 2023 at 13:03

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