For concreteness I want to work with $\mathbb P^1$. We know already that its cotangent bundle is $\mathcal O(-2)$, and the canonical compactification is the space $\mathbb{P}(\mathcal O(-2) \oplus \mathcal O)$. I've read the posts here and here, although my algebraic geometry is not good enough to understand them entirely.
I want to understand how those posts relate to the one point compactification of $T^*\mathbb P^1$.
The way I'm currently visualising it is that $\mathbb P^1$ is the sphere, $T^* \mathbb P^1$ is the sphere with a bunch of complex planes arranged to vary algebraically across the surface (their duals technically, but in dimension 1 I'm not missing anything by considering $\mathbb C^*$ as $\mathbb C$), and so $T^* \mathbb P^1$ is the same bundle but with all these complex planes compactified to the same point.
In essence, it looks like a sphere ($\mathbb P^1$) covered by a bunch of spheres (which look like $\mathbb P^1$) all glued/pulled to a point at infinity (each of the cotangent spaces at the points on $\mathbb P^1$).
It's not fully clear to me how this relates to $\mathbb{P}(\mathcal O(-2) \oplus \mathcal O)$.
In some sense, $\mathcal O(-2) \oplus \mathcal O$ is just $\mathbb P^1$ with a space that looks like $\mathbb{C}^2$ at each point instead of $\mathbb{C}$.
Then the projectification is just compactifying each of these $\mathbb{C}^2$, i.e. making $\mathbb{P}^1$, and doing so 'compatibly'.
My questions are: is the construction I outlined above 'the same' process as projectifying $\mathcal O(-2) \oplus \mathcal O$, and if so, why? Is there some 'deeper' reason/philosophy behind why they're the same process?
Best Answer
No, the one-point compactification of $T^*\Bbb P^1$ is very different from taking $\Bbb P(\mathcal{O}\oplus\mathcal{O}(-2))$. The latter is a $\Bbb P^1$ bundle over $\Bbb P^1$, while the former cannot be a fiber bundle over $\Bbb P^1$: where would you map your one added point to? One-point compactifications are not often considered in algebraic geometry - they're more of a tool of topology.