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
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2 Answers
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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.
answered Oct 11, 2012 at 16:10
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$\begingroup$ 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. $\endgroup$– HNuerCommented Oct 11, 2012 at 19:11
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$\begingroup$ 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. $\endgroup$ Commented Oct 12, 2012 at 2:39
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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.