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Let $X$ be a smooth projective scheme, $s_{1}$, $s_{2}$ global sections in $X$ and $\pi : \widetilde{X} \longrightarrow X$ be the blow up morphism of $X$ along $Y \subset X$ with exceptional divisor $E$.

Let $\widetilde{U} = \widetilde{X}\setminus E$ and suppose that $\overline{U}= \widetilde{X}$. If $s_{1} = s_{2}$ in $\widetilde{U}$ then $s_{1} = s_{2}$ in $\widetilde{X}$?

References and suggestions are welcome.

Thank you very much.

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1 Answer 1

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You need to precise what does "a section" mean to you. I guess that you are talking of a section of the structure sheaves $\mathcal{O}_X$, $\mathcal{O}_{\widetilde{X}}$.

Note that for locally free sheaves (as the structure sheaves), the zero set of a section is closed. You can apply this to $s_1-s_2$ to obtain the result you need.

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    $\begingroup$ Hi, @Dominique. Yes, I was talking about a section of the structural sheaf. Thanks for the help, thank you so much. I am new in algebraic geometry. $\endgroup$
    – Emanuell
    Dec 2, 2019 at 22:43
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    $\begingroup$ Me and my fellow beginners in algebraic geometry appreciate your response. Again, thank you very much. $\endgroup$
    – Emanuell
    Dec 2, 2019 at 22:52

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