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Is the local fundamental group of an elliptic singularity virtually solvable ? Here (the terminology is sometimes divergent) an elliptic singularity is a (germ of) normal surface $(X,x)$ such that $X$ is Gorenstein ($K_X$ is Cartier) and $R^1\pi_* \mathcal{O}_Y=\mathbb{C}_x$ where $\pi:Y\to X$ is a resolution of the singularity.

Equivalently: $\pi_*\omega_Y=m_x\omega_X$.

Thanks in advance, Benoit

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  • $\begingroup$ To motivate my question, recall that the rational Gorenstein singularities of surfaces are exactly the ADE singularities and in this case the local fundamental group is finite. $\endgroup$
    – Benoît
    Commented Nov 3, 2012 at 9:00

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Well it is not always the case: elliptic singularities of surfaces can have very complicated local fundamental group. For instance, the following example $$x^2+y^3+z^7=0$$ has a finite index subgroup which the fundamental group of a curve of genus 3 curve. To see this, note that it can be realized as a finite quotient of the cone over the Kein quartic $C$. This curve has the biggest automorphism group among curves of genus 3 ($\mathrm{Aut}(C)$ has order 168) and it acts naturally on the cone over $C$. This description can be found in some old papers of Dolgachev "Conic quotient singularities of complex surfaces" http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=58860&vfpref=html&r=72&mx-pid=345974 (see also some unpublished paper of Miles Reid).

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