In general, how to prove a variety has rational singularities intrinsically? i.e., don't use the Artin's criterion concerning the exceptional locus. And what kinds of varieties have only rational singularities? I've read a proof of classification of normal cubic surfaces (paper of Bruce and Wall), but their method is quite concrete, is there any way to prove the fact that a normal cubic surface which is not a cone can only contain rational singularities without using the classification of du Val singularities?
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2$\begingroup$ The singularity of a cubic cone is not rational! $\endgroup$– SashaCommented May 7, 2014 at 14:10
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$\begingroup$ I'm sorry! I mean normal cubic surfaces which are not cones. $\endgroup$– user50489Commented May 7, 2014 at 14:23
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For a cubic surface $X$ this is easy: for any resolution $f:\hat{X}\rightarrow X$ the surface $\hat{X}$ must be rational (the singular points of $X$ are double points since cones are excluded; project from a double point). Thus $H^i(\mathcal{O})$ vanish for $X$ and $\hat{X}$, and $i=1,2$. From the Leray spectral sequence for $f$ one gets $R^1f_*\mathcal{O}_{\hat{X}}=0$, which is one of the characterizations of rational singularities.
It is hard to give a precise answer in the general case: it depends what you know about your variety and its resolution.
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$\begingroup$ Thanks! BTW, can you recommend some references about surface singularities? $\endgroup$ Commented May 7, 2014 at 15:27
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$\begingroup$ Vast program! You might try these notes. $\endgroup$– abxCommented May 7, 2014 at 16:08