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Let $X$ be a weak del pezzo surface, which means that the anticanonical bundle $-K_X$ is nef. There is a classification of such surfaces by the configurations of its $(-2)$-exceptional components. I wonder whether there exists some classification of rational surface which is non weak del pezzo,i.e: the surface might admit $(-n)$-curves such that $n\geq 3$. For example, consider a surface $Y$ such that $(-K_Y).C\geq -1$, did anybody do some work to classify them? I expect some work like classifying them by looking at different configurations of $(-2),(-3)$-curves just like the case in weak del pezzo surfaces.

Thanks

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    $\begingroup$ This is the same as asking for a classification of all rational surfaces. Every (smooth) rational surface is a blowup of $\mathbf P^2$ or a Hirzebruch surface. I don't think you can say anything more. $\endgroup$ – potentially dense Jan 12 '17 at 11:44
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    $\begingroup$ Actually the usual definition of a weak del Pezzo surface stipulates that $-K_X$ is both nef and big. Note that rational surfaces with $-K_X$ big have not been classified and there are many partial results in the literature on a classification, see e.g. arxiv.org/abs/0901.1094. $\endgroup$ – Daniel Loughran Jan 12 '17 at 11:46
  • $\begingroup$ @DanielLoughran, thanks very much, this paper is exactly what I am looking for. I just want some more general class of rational surface than weak del pezzo whose classification was done. I need to test some statement on such class surfaces. $\endgroup$ – user393754 Jan 12 '17 at 14:08

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