On minimal resolution of singularities and the type of singularities - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:52:10Z http://mathoverflow.net/feeds/question/21010 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/21010/on-minimal-resolution-of-singularities-and-the-type-of-singularities On minimal resolution of singularities and the type of singularities Amira 2010-04-11T14:16:14Z 2010-12-15T10:00:04Z <p>Let $Y$ be a normal projective surface, let $X$ be a smooth projective surface and let $\pi:Y\longrightarrow X$ be a finite morphism. Why are all singularities of $Y$ cyclic quotient singularities? And what does this mean? Furthermore, why are these singularities rational? And again, what does that mean? (I edited the question. So the example of Ekedahl doesn't work anymore.)</p> <p>Take a minimal resolution of singularities $\rho:Y^\prime\longrightarrow Y$. Then the above apparently shows that <code>$R^0 \rho_\ast \mathcal{O}_{Y^\prime} = \mathcal{O}_Y$ and $R^i \rho_\ast \mathcal{O}_{Y^\prime} = 0$</code> for $i>0$. Is this something special for surfaces?</p> <p>The reason I ask this question is the following.</p> <p>If $Y$ is a normal variety (say over the field of complex numbers) with the above data, do we still have <code>$R^0 \rho_\ast \mathcal{O}_{Y^\prime} = \mathcal{O}_Y$ and $R^i \rho_\ast \mathcal{O}_{Y^\prime} = 0$</code> for $i>0$.</p> <p><strong>Note</strong>. The case of a surface over the complex numbers is dealt with in <em>Compact complex surfaces</em> by Barth, Hulek, Peters and van de Ven. I believe they show that cyclic quotient singularities are rational in this case.</p> http://mathoverflow.net/questions/21010/on-minimal-resolution-of-singularities-and-the-type-of-singularities/29470#29470 Answer by Richard Montgomery for On minimal resolution of singularities and the type of singularities Richard Montgomery 2010-06-25T05:31:03Z 2010-06-25T07:21:43Z <p>Re. cyclic quotient singularity. See Kollar's book: Resolution of Singularities' book. p. 81, item (3) and explanations that follow.</p> <p>I also found Durfee. L'enseignement Math. 1979. Tome 25. fasc. 1-2. p. 131. Fifteen characterizations of Rational Double Points' helpful in getting myself oriented with examples regarding isolated singularities for surfaces. </p> http://mathoverflow.net/questions/21010/on-minimal-resolution-of-singularities-and-the-type-of-singularities/40902#40902 Answer by Sándor Kovács for On minimal resolution of singularities and the type of singularities Sándor Kovács 2010-10-03T03:56:39Z 2010-12-15T10:00:04Z <p>1) You probably meant $\pi: X\to Y$ and not $\pi:Y\to X$. That way, any singularity that appears on a scheme of finite type over a field can be mapped to a smooth variety in a finite way. (This claim is implicit in VA's and Torsten Ekedahl's comments above).</p> <p>2) A fairly general criterion for a singularity to be rational is given in my paper <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.dmj/1092749293" rel="nofollow">A characterization of rational singularities</a>. In particular it covers your case in any dimension.</p>