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Nine different types of singularities are possible on a cubic surface, according to Wikipedia. How exactly is the "type" of singularity defined? I know that the number corresponding to the singularity is the number of degrees of freedom removed, but how can we say that the two different surfaces with a $D_4$ singularity have the same type of singularity while the surface with an $A_4$ singularity has another type? What properties can tell the difference between the two singularities?

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    $\begingroup$ If you are over $\Bbb{C}$: two singularities have the same type if they are analytically isomorphic. $\endgroup$
    – abx
    Commented Nov 27, 2022 at 19:34
  • $\begingroup$ Does "analytically isomorphic" mean that we can make a continuous transformation between the two types? $\endgroup$
    – mathlander
    Commented Nov 27, 2022 at 19:59
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    $\begingroup$ Analytic isomorphism does not mean algebraic isomorphism, just as a caution. $\endgroup$
    – Mohan
    Commented Nov 27, 2022 at 20:59

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All the singularities involved in this classification are Rational Double Points. These singularities are taut, in other words, their analytic type is uniquely determined by the configuration of curves in their minimal resolution.

Such a resolution is a finite set of (-2)-curves whose dual diagram is a Dynkin diagram of the same type of the singularity.

In particular, since the Dynkin diagram of type $A_4$ is non-isomorphic to the Dynkin diagram of tipe $D_4$, this allows you to distinguish the two singularities.

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  • $\begingroup$ Is triality related to the fact that there are two distinct cubic surfaces with singularity $D_4?$ $\endgroup$
    – mathlander
    Commented Nov 28, 2022 at 17:37
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A reference is the following paper:

J.W. Bruce and C.T.C. Wall. "On the classification of cubic surfaces", J. London Math. Soc. 19 (1979) no. 2, 245–256, https://doi.org/10.1112/jlms/s2-19.2.245

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  • $\begingroup$ What does this paper prove? $\endgroup$
    – mathlander
    Commented Nov 28, 2022 at 22:37
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    $\begingroup$ @StackExchangeuser the paper provides a modern-ish account of the classification of singularities on the cubic surface. $\endgroup$
    – wgabrielong
    Commented Nov 29, 2022 at 17:12

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