Normal forms of ADE singularities

Question

Given a surface $X:f(x,y,z)=0\subset \mathbb{A}^{3}_{\mathbb{C}}$ with only ADE singularities, how does one determine the correct singularity type of $X$ by computing the normal forms?

Does a similar procedure also exist in positive characteristic?

Answer 1

This may not be exactly what you are asking about, but what follows has some references. I'm guessing you are already aware of:

• V. I. Arnolʹd, Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (1975), no. 5(185), 3–65.

There are a couple different ways. I think you are indicating you can just do changes of coordinates to get to the form you want. That works, and it's not too bad. However, I think for many cases, doing resolutions of singularities and computing the dual graph is frequently faster.

See for instance:

• M. Artin, On isolated rational singularities of surfaces, American Journal of Mathematics, Vol. 88, No. 1 (Jan., 1966), pp. 129-136.

Indeed, this procedure works even in positive characteristic. However, in characteristics 2,3,5 there are different types of the "same" ADE singularity (ie, ADE singularities with the same looking resolutions but different analytic isomorphism types).

• Artin, M. Coverings of the rational double points in characteristic p. Complex analysis and algebraic geometry, pp. 11–22. Iwanami Shoten, Tokyo, 1977.
• Greuel, G.-M.; Kröning, H. Simple singularities in positive characteristic. Math. Z. 203 (1990), no. 2, 339–354.

Such characterizations are also typically possible in mixed characteristic (ie, if you have a 2-dimensional Gorenstein local ring of multiplicity 2 whose residue field is char. $p > 0$ but whose fraction field is char. $0$). As before, in many cases you can just do change of coordinates to figure out the form, and indeed that's what Artin was doing above to write down the forms. Some other references which do some of these explicit changes in coordinates in mixed characteristic include the following (Lipman's paper predates Artin's characteristic $p > 0$ paper, and he did the E8 case quite explicitly if I recall correctly).:

• J. Lipman, Rational singularities, Publ. Math. IHES 36 (1969), 195-280.

• Carvajal-Rojas, L. Ma, T. Polstra, -, K. Tucker. Covers of rational double points in mixed characteristic. J. Singul. 23 (2021), 127–150.

Note, in mixed characteristic, there's no way $\mathbb{Z}_p[x,y]/(p^2 + x^2 + y^3)$ is isomorphic to $\mathbb{Z}_p[x,y]/(x^2 + y^2 + p^3)$. The ring element $p$ is special.