Suppose we have a projective 4fold hypersurface $X\subset P^n$ with ordinary singularities along a smooth curve $C$, and suppose that there exist a projective small resolution $s:Y\to X$. let us denote by $b:Z\to X$ the "big" resolution obtained by blowing-up $C$. is it true that $b$ factors through $s$?
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$\begingroup$ I'm pretty sure this won't be true in general. (Of course, blowing up $C$ might not yield a resolution, but even if it does...). I don't know an example off the top of my head though. $\endgroup$– Karl SchwedeCommented Jan 23, 2014 at 15:56
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$\begingroup$ Let us supppose it does, just to start with an easier hypothesis. $\endgroup$– IMeasyCommented Jan 23, 2014 at 16:50
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$\begingroup$ Does "ordinary singularities" mean "ordinary double point singularities"? If so, precisely what do you mean by "ordinary singularities along $C$"? There is a notion of a "conical stratification" studied by De Concini, MacPherson, Procesi, etc. This is one way to make this notion precise. For a conical stratification where the "transverse cone" along $C$ is an ordinary double point, I expect that there is a factorization. $\endgroup$– Jason StarrCommented Jan 23, 2014 at 22:23
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1$\begingroup$ @Sandor I think with more assumptions it is easy to prove --- assume that in the small resolution $s:Y \to X$ the scheme-theoretic preimage of $C$ is a smooth surface. Then blowing up this surface you will get a smooth variety $Z'$ such that the scheme-theoretic preimage of $C$ is a Cartier divisor. By the universal property of a blowup it will give a map $Z' \to Z$. Since both are smooth and have relative Picard number 1 over $X$ it is an isomorphism. What is wrong here? $\endgroup$– SashaCommented Jan 24, 2014 at 8:40
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1$\begingroup$ @Sasha: yes, if that is the case (which is sort of what Jason suggested) then it is OK. I assumed that the essential point is whether this is true for any small resolution. If the small resolution is obtained by blowing up an ideal with a single prime component, then it is easy. Where I stopped writing up was worrying about embedded components. Also, it seems that the natural statement should be not limited to this dimension and blowing up smooth centers. I started trying to minimize the assumptions, then I realized that while I believe there is a reasonable statement, making into a ... $\endgroup$– Sándor KovácsCommented Jan 25, 2014 at 1:04
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