Let $X$ be a projective variety over $\mathbf{C}$ of dimension $n$. Let $\pi: Y \to X$ be the blow-up of a smooth point $x \in X$. Is there a nice formula for the intersection number $E^n$?
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Sure, note that $E$ is simply $\mathbb{P}^{n-1}$ and we know that the normal bundle $\mathcal{O}_E(E) \cong \mathcal{O}_{\mathbb{P}^{n-1}}(-1)$ as is well-known. It thus follows that $$(E^n)_Y = (\mathcal{O}_{\mathbb{P}^{n-1}}(-1) \cdot \ldots \cdot \mathcal{O}_{\mathbb{P}^{n-1}}(-1))_E = (-1)^{n-1}$$
