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!



Self-intersection of exceptional divisor from blowing up a subscheme
I'm afraid it does not work the way you are hoping. At least not right out of the box. First of all, why do you think that $E^2$ is even defined? Also, what exactly is $E$? Is it the pre-image of $Z$, or the reduced exceptional divisor? I assume you are thinking of the former, because the latter could be rather tricky...


Furthermore, the proof of Hartshorne II Theorem 8.24 **does not** suit blowing up an arbitrary ideal sheaf. At best it requires the ideal sheaf to be locally generated by a regular sequence. In other words, it would require $Z$ to be a local complete intersection in $X$. OK, so assume that. Then if you blow up $Z$, then the proof works, and gives you $\mathscr O(1)$, but then the question is: This is $\mathscr O(1)$ of **what**? If you look at the proof, it is $\mathscr O\_{\mathbb P(\mathscr I/\mathscr I^2)}(1)$. However, since you blew up a fat point, $\mathbb P(\mathscr I/\mathscr I^2)$ is not a "usual" $\mathbb P^1$. It is $\mathbb P^1\_Z$, i.e., a $\mathbb P^1$ over your fat point. In particular, it is non-reduced. So, in order to define $E^2$ you need to figure out a degree function for invertible sheaves on $\mathbb P^1\_Z$. I suppose one possibility is to multiply the degree function of $\mathbb P^1$ by the length of $Z$. If you do that, then you actually get what you want (assuming that $P$ was a non-singular point of $X$), but this $\widetilde X$ is not going to be even normal, so you can't expect a very good intersection theory. Perhaps you would be better off, trying to keep track of the sheaves associated to your codimension 1 subvarieties.


Finally, you could try looking at examples. The simplest is probably blowing up the ideal $(x^2,y^2)$ in $\mathbb A^2$. That gives you a pinch point and your $E$ is the singular locus (doubled). You can try out your expectations on this example.


