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Let $X$ be a surface with two divisors $D,E$. Suppose $D,E$ intersects at $P$, denote by $Z$ the scheme-theoretic intersection. Let $\mathcal{I}$ be the ideal sheaf corresponding to $Z$. Now we blow up $X$ at $P$, $\pi: \tilde{X} \to X$, with exceptional divisor $E$.

I am curious what $\pi^*\mathcal{I}$ would be. Since $Z$ is supported only at P, $\pi^*\mathcal{I}$ should have some relation with $\mathcal{O}_{\tilde{X}}(E)$. My first guess is $\pi^*\mathcal{I} = \mathcal{O}_{\tilde{X}}(-E)^{\otimes \mu}$, where $\mu$ is the local multiplicity of $Z$ at $P$. But I feel like this may not be correct because the local equations of $D,E$ may not generated the power of maximal ideal $\mathscr{m}_P^{\mu}$ (Or say, $Z \neq \mu P$ as subschemes).

So is there a clean expression of $\pi^*\mathcal{I}$ in this case, I mean write in terms of divisors on $\tilde{X}$?

Thanks!

finiteness
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    Cool question! Could you give an example for when $Z \neq \mu P$? – Thomas Manopulo Jun 04 '23 at 12:33
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    In general $\pi^* I$ has torsion. Do you mean its image in $\mathcal{O}_{\tilde{X}}$? – Mohan Jun 04 '23 at 15:26
  • @ThomasManopulo I think what I wrote in the question makes an example. It could be in the situation that $\mathscr{m}P^{\mu} \subset \mathcal{I}_P \subset \mathscr{m}{P}^{\mu-1}$, but the containment is not an equalitiy. – finiteness Jun 05 '23 at 04:24
  • @finiteness: if you mean not $\pi^*\mathcal{I}_P$, but its image in the structure sheaf, then please, edit the question accordingly. – Sasha Jun 05 '23 at 07:36
  • @Sasha Well, I think $\pi^*\mathcal{I}$ is still my main concern, after thinking for a while. The image in the structure sheaf may help, but I still like to know the whole thing. I take back my previous comment. Sorry. – finiteness Jun 05 '23 at 14:52
  • @finiteness: did you try computing the simplest case --- the blowup of a point on $\mathbb{A}^2$? – Sasha Jun 05 '23 at 15:27
  • @Sasha Good idea, let me try this case. – finiteness Jun 06 '23 at 08:33

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