Crepant resolutions of ODP's on a 3-fold - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T22:26:20Z http://mathoverflow.net/feeds/question/99075 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/99075/crepant-resolutions-of-odps-on-a-3-fold Crepant resolutions of ODP's on a 3-fold HNuer 2012-06-08T00:26:16Z 2012-10-25T16:35:42Z <p>It seems to be a well-known fact that if we have a 3-fold $X$ with only ODP singularities (ordinary double point) and a smooth Weil divisor $E$ passing through them, then by blowing-up $X$ along $E$ we get a small crepant resolution of $X$. I was wondering if anyone knew of a quick proof of this and/or a reference where it is proved. Thanks</p> http://mathoverflow.net/questions/99075/crepant-resolutions-of-odps-on-a-3-fold/109377#109377 Answer by HNuer for Crepant resolutions of ODP's on a 3-fold HNuer 2012-10-11T13:23:58Z 2012-10-11T13:23:58Z <p>Just for the sake of saving anyone else who needs this fact, we may local analytically assume that $X\cong \text{Spec }\mathbb C[x,y,z,w]/(f(x,y,z,w))$, where $E$ is defined by $x=z=0$ and $f=xP+zQ$ for polynomials $P,Q$ such that $f$ vanishes at the origin, has vanishing derivatatives at the origin, but whose Hessian is nonsingular there. This follows sinces $E$ is smooth and passes through the ODP, which has embedding dimension 4. Taking the proper primage of $X$ under the blow-up of $\mathbb C^4$ along $E$ gives explicit equations for the blow-up of $X$ along $E$, and using the Jacobian criterion shows that the blow-up is nonsingular from non-singularity of the Hessian matrix.</p> http://mathoverflow.net/questions/99075/crepant-resolutions-of-odps-on-a-3-fold/109390#109390 Answer by Sándor Kovács for Crepant resolutions of ODP's on a 3-fold Sándor Kovács 2012-10-11T16:10:54Z 2012-10-25T16:35:42Z <p>The more general phenomenon behind this is that $E$ is not a Cartier divisor at the ODPs. (A Cartier divisor could not be smooth, because that would imply that $X$ is smooth!) The pre-image of this divisor will be Cartier and hence the blow-up is not an isomorphism (as it is when one blows up a Cartier divisor). On the other hand the divisor is locally defined by $2$ equations near the ODPs, so the fibers over these points can be at most dimension $1$. Hence they are exactly dimension $1$. In other words, the blow up is a small morphism. </p> <p>Then you need to see that the blow up is indeed smooth. You may do this by verifying that the pre-image of $E$ is smooth which should not be too hard as its pre-image ought to be just the blow up of E at the ODP points (check this by an explicit local computation). These points are smooth on $E$, so the blow-up of $E$ remains smooth. Then its a smooth Cartier divisor, so the ambient space has to be smooth along it, but it is already smooth everywhere else. </p>