Limit of a series of singularities - MathOverflow

Limit of a series of singularities

2010-02-07T13:06:57Z

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".

Answer by Deepyi choudhary for Limit of a series of singularities

2012-08-21T09:08:23Z

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)

Answer by pmath for Limit of a series of singularities

2013-01-12T03:32:47Z

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.