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Let $X$ be an anti-canonical rational surface(i.e. $-K_X$ is effective) such that $K_X^2\geq 1$. Let $D$ be a $r$-class divisor ($D^2=r, D^2+D.K_X=-2$, the latter condition can be re-interpreted as $\chi(-D)=0$) such that $r\geq 0$ and the dual graph of divisor $D$ is a tree of projective lines(with possible multiplicities of each irreducible component). Thus $D.(-K_X)=D^2+2=r+2\geq 2$

Since $-K_X$ is effective, I denoted by $-K_X=C$ regarded as irreducible or reducible curves(not necessarily reduced). Does there exist any example of $X$ besides weak del pezzo surface of $K_X^2\geq 2$ to make $H^0(O_C(D))\neq 0$?

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    $\begingroup$ I don't understand how "then it works" can be an answer to "what can we say about $X$?" Can you clarify exactly what the question is? $\endgroup$
    – Lazzaro Campeotti
    Commented Apr 5, 2017 at 12:12
  • $\begingroup$ @potentiall dense, edited . If $X$ is say weak del pezzo surface of degree $K_X^2\geq 2$, then $|-K_X|$ is base point free, then the generic member of this linear system is smooth elliptic curve, in this case $H^0(O_C(D))\neq 0$. $\endgroup$
    – user41650
    Commented Apr 5, 2017 at 13:29

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