It seems to be a wellknown fact that if we have a 3fold $X$ with only ODP singularities (ordinary double point) and a smooth Weil divisor $E$ passing through them, then by blowingup $X$ 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 preimage of this divisor will be Cartier and hence the blowup 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 preimage of $E$ is smooth which should not be too hard as its preimage 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 blowup 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. 


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 blowup of $\mathbb C^4$ along $E$ gives explicit equations for the blowup of $X$ along $E$, and using the Jacobian criterion shows that the blowup is nonsingular from nonsingularity of the Hessian matrix. 

