I have the total space of $\mathcal{O}_{\mathbb{P}^1} (-2)$ and I see that a "standard" way to compactify is to add the trivial line bundle, $\mathcal{O}_{\mathbb{P}^1}$, and then projectivize. That is $$ \mathcal{O}_{\mathbb{P}^1} (-2) \rightarrow \mathbb{P}\left(\mathcal{O}_{\mathbb{P}^1} (-2) \oplus \mathcal{O}_{\mathbb{P}^1}\right). $$

Can someone help me understand

1) Why this is a standard way to compactify something, and

2) Why do we add the trivial bundle $\mathcal{O}_{\mathbb{P}^1}$? Thanks for the help.


a) Given a field $k$, the standard way to projectify the vector space $k^n$ is to consider the projective space $\mathbb P^n(k)$ together with the embedding $ k^n \hookrightarrow \mathbb P^n(k)$ defined by $j(a_1,\dots,a_n)=[a_1:\dots:a_n:1] $, so that $k^n$ can be identified with the complement of the hyperplane $x_{n+1}=0$ in $\mathbb P^n(k)$.

b) This is all well and good but not canonical: a $k$-vector space $V$ of dimension $n$ is not $k^n$ in general.
The canonical projectification of $V$ is the projective space $\mathbb P(V\oplus k)$ together with the injective map map $j:V\hookrightarrow \mathbb P(V\oplus k):v\mapsto [(v,1)] $, which identifies $V$ with the complement of the hyperplane $\mathbb P(V\oplus 0)\subset \mathbb P(V\oplus k)$.

c) The road to the generalization to vector bundles is now clear: a vector bundle $\mathcal V$ over a geometric space $X$ (like a topological or differentioal manifold, an analytic space, an algebraic variety,...) is a collection of vector spaces $V(x), x \in X$ varying in a suitable way (continuously, diffferentiably, analytically, algebraically,...) with $x$.
The projectification of $\mathcal V$ is thus quite naturally $\mathbb P(\mathcal V\oplus k_X)$, where $k_X$ is the trivial line bundle over $X$.
This is a bundle over $X$ whose fibre at $x$ is the projective space $\mathbb P(V(x)\oplus k)$.
[But remember that in algebraic geometry the line bundle $k_X$ is notationally identified with the structural sheaf $\mathcal O_X$]
Your situation is an example of the general construction above.

  • $\begingroup$ Part (b) is very nice. $\endgroup$ – Jake Levinson Aug 1 '15 at 23:16
  • $\begingroup$ Thanks so much! Makes sense. $\endgroup$ – user46348 Aug 2 '15 at 18:11
  • $\begingroup$ You're welcome, dear user. $\endgroup$ – Georges Elencwajg Aug 2 '15 at 18:15

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