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Let $X=\operatorname{Proj}(R)$ be a projective scheme with a $k$-algebra $R$. Let $\mathcal{M} \in \operatorname{Coh}(X)$ be a coherent sheaf on $X$ whose Hilbert polynomial $\chi(\mathcal{M})$ is a constant integer $n>0$. My questions are following;

  1. Is it true that $\mathcal{M}$ is necessarily of the form $\mathcal{O}_Z$ for some subscheme $Z\subset X$ consisting of "$n$ points" (taking multiplicity into account)?
  2. Is it true that $H^0(X,\mathcal{M}(d))=n$ for all $d\ge0$, i.e. the Hilbert polynomial stabilizes in the very beginning?

Motivation: There is an equivalence of categories $$ \operatorname{Coh}(X)\cong \operatorname{gr}(A)/\operatorname{tor}(A), $$ where $\operatorname{gr}(A)$ is the categories of f. g. graded modules and $\operatorname{tor}(A)$ is its subcategory consisting of torsion modules. I want to characterize the module $\mathcal{M}$ above in $\operatorname{gr}(A)/\operatorname{tor}(A)$. A naive guess is that a module $M \in \operatorname{gr}(A)$ with $\dim_k M_d=n$ that are generated in degree $0$ yields such $\mathcal{M} \in \operatorname{Coh}(X)$. I would appreciate any help or comments.

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