Blowing up

Remarks

The ideal sheaf $\pi^{-1}\mathcal{I} \cdot \mathcal{O}_{\operatorname{Bl}_Z X}$ of $E$ is invertible, so $E$ is an effective Cartier divisor. The morphism $\pi$ is projective, and it is an isomorphism over $X \setminus Z$. It has a universal property: every morphism $f \colon Y \to X$ such that $f^{-1}\mathcal{I} \cdot \mathcal{O}_Y$ is invertible factors uniquely through $\pi$.

Example: the blow-up of $\mathbb{A}^2 = \operatorname{Spec} k[x, y]$ at the origin, with $\mathcal{I} = (x, y)$, is the closed subvariety $$\set{((x, y), [u : v]) \in \mathbb{A}^2 \times \mathbb{P}^1 : x v = y u}.$$ Over a point $(x, y) \ne (0, 0)$ the condition forces $[u : v] = [x : y]$, so $\pi$ is bijective there. Over the origin every $[u : v]$ is allowed, so $E \cong \mathbb{P}^1$, the projective line of directions through the origin.

Non-example: the projection $\mathbb{A}^2 \times \mathbb{P}^1 \to \mathbb{A}^2$ is not the blow-up at the origin. It fails to be an isomorphism over $\mathbb{A}^2 \setminus \set{0}$, since every fibre is a whole $\mathbb{P}^1$.

Blowing up an effective Cartier divisor changes nothing: $\pi$ is then an isomorphism.