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Suppose $X$ is a log del pezzo projective surface of index $l$. As far as I understand it will have a finite number of singular points all of which can be resolved by sucessive blow-ups.

Let $E_i$ be the exceptional divisors of the minimal resolution. Their self-intersection numbers are $E_i^2\leq -2$. Is there a lower bound on these numbers?

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    $\begingroup$ I am not an expert, but, for $n \geq 2$, is the weighted projective plane $\mathbb{P}(1,1,n)$ a log del Pezzo surface? If so, then its minimal resolution is the Hirzebruch surface $\mathbb{F}_n$ and it contains a curve of self-intersection $−n$. $\endgroup$
    – M P
    Aug 5, 2011 at 12:57
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    $\begingroup$ And I think the index of $\mathbb{P}(1,1,n)$ is $n$ so it would still be bounded by the index ;) $\endgroup$
    – Jesus Martinez Garcia
    Aug 5, 2011 at 16:58

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I am not sure exactly what you mean when you say log del Pezzo, but if you have klt singularities then they are quotient singularities in dimension two. For a fixed index there is a finite number of possible groups that can occur and therefore a finite number of minimal resolutions. So there is a bound in terms of the index. You can look in Nikulin, "Del Pezzo surfaces with log-terminal singularies, I" and work out an bound: $$ 1+ \sum\left(\frac{b_i}{2}-1\right) \leq l$$ where $b_i=-E_i^2$ and $E_i$ are the curves in the minimal resolution.

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