Is there an example of a hypersurface $X$ of some projective space $\mathbb{P}^n$ such that there exists an invertible sheaf $\mathcal{L}$ on $X$, not isomorphic to the structure sheaf $\mathcal{O}_X$, but has the same Hilbert function as $\mathcal{O}_X$?
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4$\begingroup$ On a plane curve, any degree zero line bundle (and they exist if degree is at least three) has the same Hilbert polynomial as the structure sheaf, by Riemann-Roch. $\endgroup$– MohanCommented Aug 22, 2018 at 0:13
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$\begingroup$ @Mohan Sorry, I should have been a bit more specific. I will edit the question to replace Hilbert polynomial by Hilbert function. Then your example does not hold (only degree zero line bundle with non-trivial global section on a curve is the structure sheaf). $\endgroup$– JanaCommented Aug 22, 2018 at 0:35
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2$\begingroup$ If $\mathscr L$ has a non-trivial global section, then there is a non-trivial embedding $\mathscr O_X\to \mathscr L$ and if $\mathscr F=\mathscr L/\mathscr O_X$, then $h^0(\mathscr F(n))\neq 0$ while $h^i(\mathscr F(n))=0$ for $n\gg 0$ and $i>0$. So, $\chi(\mathscr O_X(n))\neq \chi(\mathscr L(n))$. $\endgroup$– Sándor KovácsCommented Aug 22, 2018 at 1:20
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