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https://downloads.hindawi.com/journals/tswj/2014/346126.pdf

In this paper, Stable sheaves on a smooth quadric surface with linear Hilbert bipolynomials(E. Ballico and S.Huh), I have a question.

On the right down of the page 2 of this paper,

For every semistable 1-dimensional sheaf $F$ with $\chi_F(x, y)=mx+ny+t$, define $C_F:=\mathrm {Supp}(F)$ to be its scheme-theoretic support and then we have $C_F \in |O_Q(n, m)|$.

Question : In this sentence, how we conclude that $C_F \in |O_Q(n, m)|$?

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    $\begingroup$ This should follow from Riemann-Roch. Imagine that the sheaf is a rank 1 sheaf on a curve of type $(a,b)$, use RR to compute $\chi$ of its twists, and compare. So, the only problem is to show that its rank is 1 (it could be, say, a rank 2-sheaf on a curve of type $(n/2,m/2)$, or the curve could have several components and the sheaf could have different rank on different components), but this probably follows from other assumptions in the paper. $\endgroup$
    – Sasha
    Commented Jun 9, 2021 at 4:51
  • $\begingroup$ @Sasha Thank you for the comment. However, I cannot understand your explanation since my lack of comprehension. Please explain above thing more detail. $\endgroup$
    – H.S. Kim
    Commented Jun 9, 2021 at 6:22
  • $\begingroup$ What part of the explanation are you missing? Can you apply RR? $\endgroup$
    – Sasha
    Commented Jun 9, 2021 at 6:42
  • $\begingroup$ I know RR. But how apply RR to compute $\chi$ of its twist? $\endgroup$
    – H.S. Kim
    Commented Jun 9, 2021 at 6:47
  • $\begingroup$ Do you know how a line bundle twist modifies the Chern character of a sheaf? $\endgroup$
    – Sasha
    Commented Jun 9, 2021 at 7:15

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