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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?

flag	asked Nov 9 at 8:38 Zheng 21●1

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. – temp Nov 9 at 16:02

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@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 $\mathbb{P}^1$ . – Jason Starr Nov 9 at 16:06

Jason Starr · Answer 1 · 2012-11-09 16:02:44Z UTC

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

answered Nov 9 at 16:02

Jason Starr

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Some bounded theorem of algebraic stack of coherent sheaves

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