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Let $X$ be a Noetherian scheme. My question is that: for any Cartier divisor $D$, can we write it as $D_1-D_2$ where $D_1$,$D_2$ are effective? What about further assume $X$ is integral?

I can see $D$ is locally generated by fractions of sections of $O_X$. But it seems not easy to depart the denominators and numerators explicitly. Could you provide some help? Thanks.

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  • $\begingroup$ Related, but unanswered: math.stackexchange.com/questions/3992498/… $\endgroup$
    – KReiser
    Jan 22, 2022 at 8:25
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    $\begingroup$ If $X$ has an ample divisor, the answer is positive. $\endgroup$
    – Sasha
    Jan 22, 2022 at 8:47
  • $\begingroup$ @Sasha Do you mean $X$ has an ample invertible sheaf? And could you give a proof or share a reference? Thanks! $\endgroup$
    – Richard
    Jan 22, 2022 at 12:52
  • $\begingroup$ @Richard: This can be found in Hartshorne, but you can also see a simple argument below. $\endgroup$
    – Sasha
    Jan 22, 2022 at 18:33
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    $\begingroup$ @Richard: this may also help stacks.math.columbia.edu/tag/0AYM $\endgroup$
    – Ben C
    Jan 22, 2022 at 20:56

1 Answer 1

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Assume $X$ has an ample (Cartier) divisor class $H$ (equivalently, an ample invertible sheaf). Then for each Cartier divisor $D$ there exists an integer $n \gg 0$ such that the linear systems $|nH|$ and $|D + nH|$ contain effective (Cartier) divisors, say $D_1$ and $D_2$. Then $$ D_2 - D_1 = (D + nH) - nH = D. $$

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  • $\begingroup$ I can't understand how to deduce these two linear systems contain effective Cartier divisors. Could you provide a reference? Thanks! $\endgroup$
    – Richard
    Jan 23, 2022 at 14:39
  • $\begingroup$ Read the part about ample divisors in the Hartshorne textbook. $\endgroup$
    – Sasha
    Jan 23, 2022 at 14:43

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