# Three-dimensional analogues of Hirzebruch surfaces

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 ?

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

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.

• Thank you for this very nice answer. Nov 29, 2023 at 13:03