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Let $X \subset \mathbb{C}^n$ be a complex complete intersection surface with only ordinary double point singularities. Let $o$ be such an ordinary double point.

Let $\tilde{X}$ be the strict transform of $X$ with respect to the blow up of $\mathbb{C}^n$ at $o$. My question is, how does $\tilde{X}$ look like?

I know that locally at $o$, the strict transform should look like a a smooth rational curve with self-intersection $−2$, and should look like $X$ far away from $o$. (This can be formalized by taking a small neighborhood of $o$). But how do these two parts "glue"?

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    $\begingroup$ This is incorrect --- locally at $o$ the strict transform looks like a neighborhood of a smooth rational curve with self-intersection $-2$. $\endgroup$
    – Sasha
    Commented Mar 29, 2023 at 17:49
  • $\begingroup$ @Sasha why? Locally at $o$, $X$ would be a sum of squares by Morse's lemma. $\endgroup$
    – Serge the Toaster
    Commented Mar 29, 2023 at 19:45
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    $\begingroup$ It is a sum of three squares. $\endgroup$
    – Sasha
    Commented Mar 30, 2023 at 4:52
  • $\begingroup$ @Sasha You are right. I corrected the question accordingly. Thanks! $\endgroup$
    – Serge the Toaster
    Commented Apr 6, 2023 at 17:44

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