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Let $X$ be a smooth weak fano threesfold arising as the blowup of a smooth curve $C$ of degree and genus $(d,g)=(10,2)$ in a rank 1 smooth fano threefold $Y$, $-K_Y^3 = 22$. Let $H$ be a hyperplane in $X$ and $E$ the exceptional divisor. What is $h^0(X, 4H-5E)$?

I think it can be 1 or 3, but not sure how to compute it directly.

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  • $\begingroup$ It depends on which H you choose. $\endgroup$
    – Ennio Mori cone
    Commented Oct 12 at 11:16
  • $\begingroup$ H here is the pullback of -K_X. $\endgroup$
    – user15720
    Commented Oct 13 at 13:15

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