Minimal resolution of Log del Pezzo surfaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T14:52:59Z http://mathoverflow.net/feeds/question/72083 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/72083/minimal-resolution-of-log-del-pezzo-surfaces Minimal resolution of Log del Pezzo surfaces Jesus Martinez Garcia 2011-08-04T12:52:21Z 2013-01-29T01:22:00Z <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/72083/minimal-resolution-of-log-del-pezzo-surfaces/118928#118928 Answer by Colin Ingalls for Minimal resolution of Log del Pezzo surfaces Colin Ingalls 2013-01-14T22:58:24Z 2013-01-14T22:58:24Z <p>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.</p>