Let $X$ be a surface $P$ be a point on $X$. Let $Z$ be a subscheme supported on the single point $P$ ($Z$ not equal $P$), denote its local multiplicity by $\mu_P(Z)$.

Now blow up $\pi: \tilde{X} \to X$ along $Z$, we get corresponding exceptional divisor $E$. I would like to know what is $E^2$. I made my guess that it might be $-\mu_P(Z)$ but I am not sure.

I tried to do the same calculation as how monoidal transform being done in Hartshorne. The self intersection is defined to be (Hartshorne V 1.4.1) $E^2=\deg_E \mathcal{N}_{E/\tilde{X}}$, where $\mathcal{N}_{E/\tilde{X}}$ is the normal sheaf. From Hartshorne II Theorem 8.24, which suits for any blow up along an ideal sheaf, the normal sheaf is isomorphic to $\mathcal{O}_{E}(-1)$. So $E^2$ looks still like $-1$.

But clearly this blow up is not monoidal, by Castelnuovo contract criterion, $E^2$ should not be $-1$. I am confused here.

Any comment is appreciated! If there is any reference for this situation would be great!