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Let $C$ be a curve lying on a smooth surface $X$ in $\mathbb P^3$. Let $P$ be a singular point of $C$ and $m$ is the multiplicity of $C$ at $p$. If we consider the blow - up $\pi : B \to X$ of $X$ at $p$ and denote $ C'$ to be the strict transform of $C$ by $\pi $ and $E$ to be the exceptional divisor, then is the following true : $ \pi ^*C = C' + m E$ ?

Can anyone give me any reference on this.

Any help from anyone is welcome

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    $\begingroup$ Prop 3.6 of chapter V of Hartshorne $\endgroup$
    – peter a g
    Jan 27, 2020 at 13:36
  • $\begingroup$ @peterag,thanks. $\endgroup$
    – HARRY
    Jan 27, 2020 at 13:38

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This is a community wiki post consisting of the answer from the comments, in order to remove this question from the unanswered queue.

Prop 3.6 of chapter V of Hartshorne – peter a g

Here is the statement of the proposition (with some clarifying language on the assumptions):

Proposition 3.6. Let $X$ be a smooth surface. Let $C$ be an effective divisor on $X$, let $P$ be a point of multiplicity $r$ on $C$, and let $\pi:\widetilde{X}\to X$ be the monoidal transformation with center $P$ (the blowup of $X$ at $P$). Then $$ \pi^*C = \widetilde{C} + rE$$ where $\widetilde{C}$ is the strict transform of $C$.

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  • $\begingroup$ Thanks, especially for completing my 'answer.' Furthermore, one really should not leave answers in comments (as I did), no matter how short they are, if there is a chance that the result is the question remaining in the unanswered queue. $\endgroup$
    – peter a g
    Jan 28, 2020 at 3:19
  • $\begingroup$ Happy to do it, happy to remove this if you're going to post one yourself. (I don't think we need to be quite that slavish in our devotion to the unanswered queue, but if you know what you're posting is an answer, I agree that one should post it as an answer.) $\endgroup$
    – KReiser
    Jan 28, 2020 at 3:54
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    $\begingroup$ No - I'm just as happy to leave as it is (wiki), esp. with the extra details. As I'm sure you know, the urge to leave an answer in the comments stems from feeling that one's contribution actually = 0. So community wiki is perfect. $\endgroup$
    – peter a g
    Jan 28, 2020 at 3:58

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