# singularities preserved by integral closure

Let $X$ be an affine variety. Let $A$ be the coordinate ring of $X$ and let $K$ be the fraction field of $A$. Given a Galois extension $K\subset L$, let $B$ be the integral closure of $A$ in $L$. Let $Y$ be the associated affine variety of $B$.

Questions: can rational (Gorenstein) singularities be inherited from $X$ to $Y$?

• Generally there is no hope that things like this can be true. Under some conditions (if $Y \to X$ is etale in codimension 1 for example) then if $X$ is log terminal then so is $Y$. This kind of statement does not hold for rational singularities though. Although in the Gorenstein case, rational singularities are equivalent to log terminal singularities... (and log terminal always implies rational). – Karl Schwede Jul 12 '14 at 3:46
• Thanks. Could you give some relevant references? I'm not quite aware of some standard AG terminology, what do you mean by etale in codimension 1? Maybe you mean etale outside a codimension 2 subvariety? – JJH Jul 12 '14 at 3:53
• Yup, I mean etale outside a codimension 2 subvariety. Perhaps the right reference is Kollar-Mori for the statement I cited. For rational singularities though this is false, see for instance the work of Anurag K. Singh on canonical covers of rational singularities. – Karl Schwede Jul 12 '14 at 20:06