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Let $X$ be a connected projective scheme. Let $U$ be a finite type open substack of the algebraic stack of coherent sheaves on $X$ with a fixed Hilbert polynomial. Can one take $p>0$ such that every sheaf in $U$ is Castelnuovo-Mumford $p$-regular? It seems true and used in some papers (maybe with additional condition), but I cannot show this statement. Could anyone give me a sketch of a proof?

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  • $\begingroup$ See FGA Explained, or Mumford's book on curves on algebraic surfaces. I think you can obtain a p only depend on the hilbert polynomial. $\endgroup$
    – temp
    Commented Nov 9, 2012 at 16:02
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    $\begingroup$ @temp: There is no $p$ that depends only on the Hilbert polynomial. To see this, consider the infinite sequence of locally free sheaves $\mathcal{O}(-n)\oplus \mathcal{O}(n)$ on $\mathbb{P}^1$. $\endgroup$
    – Jason Starr
    Commented Nov 9, 2012 at 16:06

1 Answer 1

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Lemma 1.7.6, p. 28 of the book by Huybrechts and Lehn.

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