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The $A_\infty$ and $D_\infty$ plane curve singularities have defining equations $x^2=0$ and $x^2y=0$. These equations are "clearly" natural limiting cases of the equations for $A_n$ singularities $x^2 + y^{n+1}=0$ and $D_n$ singularities $x^2y+y^{n-1}=0$ as $n \to \infty$, since large powers are small in the adic topology. So we're tempted to say that $A_\infty$ and $D_\infty$ are "limits" of the "series of singularities" $\{A_n\}$ and $\{D_n\}$. This was already observed by Arnol'd in 1981, who wrote "Although the series undoubtedly exist, it is not at all clear what a series of singularities is."

Have there been any attempts since Arnol'd to make sense out of the phrases in quotes in the previous paragraph? That is:

Are there precise definitions of a "series of singularities", and of the "limit" of a series of singularities, under which $\lim_{n\to \infty} A_n = A_\infty$ and $\lim_{n\to \infty} D_n = D_\infty$?

If the answer is Yes, here's another desideratum: does the notion of "limit" extend to modules/sheaves over the singularities? My motivation here is that the $A_n$ and $D_n$ are (almost) precisely the equicharacteristic hypersurfaces with finite Cohen-Macaulay type (i.e. only finitely many indecomposable MCM modules), while $A_\infty$ and $D_\infty$ are precisely the ones with countable or bounded CM type. I'd really like some statement that each MCM module over the "limit" "comes from" a module "at some finite stage".

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  • $\begingroup$ I think the answer might be No. The reason is that varieties/singularities are parametrized by coefficients rather than by degrees. But that is my guess which won't serve a correct/formal answer to your question. $\endgroup$
    – 7-adic
    Commented Mar 20, 2010 at 19:45
  • $\begingroup$ regarding the second point, note that in both cases you can put in a scaling parameter and get families $x^2 + \epsilon y^{n+1}$ and $x^2 y + \epsilon y^{n-1}$. Taking $\epsilon \to 0$ gives a map from matrix factorizations of $A_n$ to matrix factorizations of $A_\infty$, which is presumably injective and eventually gets everything, so I guess a posteriori you learn that everything over the limit comes from a finite stage. $\endgroup$
    – Vivek Shende
    Commented Feb 23, 2013 at 6:39
  • $\begingroup$ my previous comment was really meant to be a question: are you asking that the notion of limit should be compatible with the above described maps? $\endgroup$
    – Vivek Shende
    Commented Feb 23, 2013 at 6:43
  • $\begingroup$ You might like to read: Schrauwen "topological series of isolated plane curve singularities" $\endgroup$
    – Dmitry Kerner
    Commented Jan 13, 2014 at 19:20

2 Answers 2

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This is not an answer, but rather a long comment (grad student level, so please don't take it seriously). I use surfaces for simplicity. The answer must yes in some form. My belief is from the moduli space theory. It is known that the normal stable surfaces admit at worst log canonical isolated singularities. This includes $xyz+x^p+y^r+z^q$ singularities. However, to have a complete moduli space of surfaces, we must include no isolated singularities of the form $xyz$, $xyz+x^p$, and $xyz+x^p+y^r$ (among others). The resemblance of the equations must be more than a coincidence. So, I can imagine we can have an isolated singularity and consider all the deformations from it to non isolated ones. Then to look for the minimal "complete" family of such degenerations.

I wish someone can say something more about all this.

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Large power are small in adic topology: For series of singularities let A infinity and D infinity are plane curve singularities:x^2=0 & X^2.y=0 both equations are natural limiting for A(n) singularities x^2+y^n+1=0 & D(n) singularities x^2y+y^n-1=0 as n_>infinty thats why here we are able to say that A (infinity) and D(infinity) are limits of series of singularity A(n) and D(n)

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    $\begingroup$ Isn't this just the first paragraph in the question? $\endgroup$
    – Ketil Tveiten
    Commented Aug 21, 2012 at 9:49

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