4
$\begingroup$

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

$\endgroup$
1
  • $\begingroup$ 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
    Nov 3, 2012 at 9:00

1 Answer 1

4
$\begingroup$

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http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=58860&vfpref=html&r=72&mx-pid=345974 (see also some unpublished paper of Miles Reid).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.