# When are the fibers of a resolution of singularities reduced?

I apologize if this is too much of a fishing expedition, but I've had bad luck searching for any literature on this subject, and I was hoping someone could tell me if it's too easy to be worth mentioning, too hard to say anything about, or if I'm using the wrong key words.

Let $X\to Y$ be a resolution of singularities.

Is there any adjective I could apply to the above resolution which would assure that its scheme-theoretic fibers are reduced? Perhaps crepant, or maybe symplectic? What if the base has rational singularities?

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The usual resolutions of the D_n, E_6, E_7, and E_8 surface singularities do not have reduced fibers. The multiplicity of a component is the coefficient in the maximal root of the corresponding simple root. – David Treumann Nov 17 '11 at 2:03
So, that would be a no.... – Ben Webster Nov 17 '11 at 3:43
As David has pointed out, crepant and rational don't work (I know little about symplectic things). I think crepant is the wrong kind of condition, as it attempts to duplicate minimality in dimensions greater than 2; more plausible to me might be a statement like given singular X such that a resolution with reduced fibres exists and so does a crepant resolution, then the crepant resolution has reduced fibres'. No idea if this is true though. – David Holmes Nov 17 '11 at 14:11
I came across the same question in my masters thesis; if I could show that a certain class of surface singularities had reduced resolutions in the above sense then I could get nice results easily. However, I never got anywhere with a general condition for this, and ended up just doing lots of messy calculations to get around it in the case I needed. As such, I suspect the answer may be no', but would also be interested if anyone has a positive answer! – David Holmes Nov 17 '11 at 14:13
I agree with Treumann that it is not a property that holds for one of the classes of singularities generally used in the MMP. But you might check conical singularities in the sense of DeConcini, MacPherson and Procesi. Related, if there is a coherent sheaf such that Proj Sym*(E) is smooth, then you can cut with hyperplane sections to get a resolution with reduced fibers (even just projective spaces -- of varying dimensions -- as fibers). That does occur sometimes. – Jason Starr Nov 17 '11 at 14:26