Blowup of $\mathbb{P}^n$ at a point is irreducible 
The blowup of $\mathbb{P}^n$ at a point is irreducible.

This seems clear intuitively, but I'm not sure how to prove it. Thoughts?
 A: The isomorphism between $\mathbb{P}^n - {0}$ and the blowup without the exceptional divisor (call that set $U$) gives a continuous map from $\mathbb{P}^n - {0}$ to $U$. $\mathbb{P}^n - {0}$ is irreducible, so its image under a continuous map, $U$, is also irreducible. $U$ is dense in the blowup, the closure of a irreducible set is irreducible and thus the blowup is irreducible.
A: This has nothing to do with $\mathbb P^n$ ; the blow-up of a variety $X$ always gives a birational isomorphism $\pi : \widetilde{X} \dashrightarrow X$, hence the inverse rational map $\pi^{-1} : X \dashrightarrow \widetilde X$ is dominant (see Hartshorne Chapter I, Theorem 4.4 ; the birational map gives rise to an isomorphism $K(\widetilde X) \simeq K(X)$, hence the corresponding inverse rational map is dominant). 
This means that the image of $\pi^{-1} : X \dashrightarrow \widetilde X$ is dense, and this rational map can be represented by the inverse of the isomorphism $\pi|_{\pi^{-1}(X/Y)} : \pi^{-1}(X/Y) \to X/Y$ (this is Chapter II, Proposition 7.13 in Hartshorne). Since the image of irreducible varieties under a morphism is irreducible, we conclude that $\pi^{-1}(X/Y)$ is irreducible and dense, hence $\widetilde X$ is irreducible. 
Of course this argument requires a bit more machinery, but it's nice to know in general that blow-up preserves irreducibility. 
Hope that helps,
