# Crepant resolutions of ODP's on a 3-fold

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

-

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.
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.
Yeah, in my head the only question that was bothering me was the smoothness. It seemed clear to me that the exceptional locus would just be $\mathbb P^1$'s which wouldn't contribute to the discrepancy. As usual you have explained things very clearly. –  HNuer Oct 11 '12 at 19:11
Actually, I think that verifying that the pre-image of $E$ is smooth should not be too hard as its pre-image ought to be just the blow up of $E$ at those points, which are smooth points, so the blow-up 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. –  Sándor Kovács Oct 12 '12 at 2:39
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.