Self-intersection of exceptional divisor from blowing up a subscheme

Question

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!

Answer 1

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.