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Given an algebraic variety with a singularity is there an effective way to determine whether this singularity is or is not ADE.

I do know that this can be solved by computing the modality of the singularity and checking whether this is, or is not zero. However I do not know how to compute the modality of a singularity effectively.

I should explain my problem slightly better to reply to the questions being asked. I have a long list of surfaces with a singularity. For each of these I would like to know if these singularities are ADE or not.

One example would for instance be the surface with equation: x^2+x^5+y^5+z^5. I do not know how to determine the type of singularity at the origin.

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  • $\begingroup$ The SINGULAR software can help. $\endgroup$
    – Liviu Nicolaescu
    Apr 24, 2012 at 13:27
  • $\begingroup$ Is it a surface singularity? $\endgroup$
    – Karl Schwede
    Apr 24, 2012 at 13:29
  • $\begingroup$ The case that interests me most is when the variety is a surface. The singularity is then either an isolated point or a curve. $\endgroup$
    – Heijne
    Apr 24, 2012 at 13:55
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    $\begingroup$ ADE singularities are normal and therefore isolated, so perhaps that rules out some of your examples. They can be characterized as rational Gorenstein surface singularities. This might be checkable depending on what you know. $\endgroup$
    – Donu Arapura
    Apr 24, 2012 at 14:19
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    $\begingroup$ The example you give is not Du Val. See condition (3) of Theorem 2.1 in "The Du Val Singularities" by Miles Reid and also Section 2.2. In particular it says that the defining equation needs to have a monomial $x^i y^j z^k$ with non-negative coefficient and such that $2i + j + k < 4$. None of the terms in your equation satisfy this condition. $\endgroup$
    – Dustin Cartwright
    Apr 25, 2012 at 13:58

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