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I am studying Mumford's 'Lectures on curves on an algebraic surface', in particular the section on Castelnuovo Mumford regularity. I am stuck in a small point. A coherent sheaf $F$ on $P^n$ is $m$-regular if $H^i(P^n,F(m-i))=0$ for all $i>0$.

He says the following. If $F$ is a $m$- regular sheaf on $P^n$, then $H^0(P^n,F(k))$ is spanned by $H^0(P^n,F(k-1))\otimes H^0(P^n,O(1))$ if $k>m$.

As a consequence of the above statement and Serre's theorem ($F(k)$ is generated by it's sections if $k$ is large enough), he says for $k$ large, $F(k)$ is generated by $H^0(F(m))\otimes H^0(F(k-m))$. I am not able to make this jump. Why are we getting this as a consequence.

I would be grateful if someone clarifies, even though this might be a silly question.

user52991
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1 Answers1

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It might be something like this: a repeated application of (a) implies that $H^{0}(F(m)) \otimes (H^{0}(O(1)))^{\otimes k-m} \to H^{0}(F(k))$ is surjective; but this map factors through $H^{0}(F(m)) \otimes H^{0}(O(k-m)) \to H^{0}(F(k))$, by the natural map $(H^{0}(O(1)))^{\otimes k-m} \to H^{0}(O(k-m))$.

Minseon Shin
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