Let $X$ be a weak del pezzo surface($-K_X$ is nef and $K_X^2>0$) with $3\leq K_X^2\leq 5$. Let $H$ be a $1$-class on $X$ such that $H.C\geq 1$ for any $C\in I^{irr}(X)$. Then $2H+K_X$ is effective divisor.

A divisor $D$ is called $r$-class if $D^2+D.K_X=-2,D^2=r$, here $H$ is $1$-class means that $H^2=1,H.K_X=-3$. $I^{irr}(X)$ is set of irreducible $(-1)$-curves on $X$.

The statement above can be verified case by case since we have complete classification of weak del pezzo surfaces, but I do not know how to get a general proof.

  • $\begingroup$ Dou you consider only smooth surfaces? $\endgroup$ May 18, 2017 at 2:28
  • $\begingroup$ yes,I only consider smooth case $\endgroup$
    – user41650
    May 18, 2017 at 11:03
  • $\begingroup$ In case $X$ is a del Pezzo surface, does a 1-class divisor H exist? In any case, your question is equivalent to prove that $H$ is not nef, right? $\endgroup$ May 20, 2017 at 2:49


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