I read about Du Val singularities on surface are classified by equations of ADE type. For example, $x^2+y^2+z^{n+1}=0$ for A type. As not every surface can have a neighbourhood embedded in $\mathbb{A}^3$. Why do we use analytic coordinates to characterize singularity?
$\begingroup$
$\endgroup$
1
-
$\begingroup$ The same issue comes up when describing families of semistable curves (say with smooth generic fiber): one can always get by using the etale topology (appropriately formulated) without recourse to analytic methods, ultimately due to the Artin approximation theorem. $\endgroup$– user76758Commented Jun 9, 2014 at 19:45
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
5
The point is that, locally, any Du Val singularity (that is, any isolated surface singularity that arises by contracting an $A$-$D$-$E$ curve) can be realized as a double cover of a nonsingular surface. This means that there exist local analytic coordinates such that the germ of singularity has the form $$x^2=f(y, \, z),$$ that is the corresponding embedding dimension is $3$. In other words, even if the surface is not globally embeddable in $\mathbb{A}^3$, an analytic neighbourhood of the singularity always is.
A good reference is [Barth-Peters-Van De Ven, Compact Complex Surfaces, Springer 1984], see in particular Lemma 3.8 of Chapter III.
answered Jun 9, 2014 at 15:13
-
$\begingroup$ dear Francesco: could one use étale or formal coordinates instead? $\endgroup$ Commented Jun 9, 2014 at 15:58
-
$\begingroup$ For formal coordinates the answer is yes. This follows from Corollary 1.6 of M. Artin's paper On the solutions of analytic equations.. $\endgroup$ Commented Jun 9, 2014 at 16:22
-
$\begingroup$ @user125763: It also follows from Artin approximation (the earlier paper of Artin...) that if $(X,x)$ and $(X',x')$ are pointed schemes of finite type over a field or excellent Dedekind domain $R$ and $f:\mathscr{O}_{X,x}^{\wedge} \simeq \mathscr{O}_{X',x'}^{\wedge}$ is an $R$-isomorphism then there is a common residually trivial pointed etale neighborhood $(X'',x'')$ of $(X,x)$ and $(X',x')$ which induces an isomorphism between those completed local rings that agrees with $f$ modulo whatever power of the maximal ideals you like. Same for arbitrary excellent schemes via Popescu's approx. thm. $\endgroup$ Commented Jun 9, 2014 at 19:43
-
$\begingroup$ @user76758: thanks! can I ask what residually trivial mean? $\endgroup$ Commented Jun 9, 2014 at 20:20
-
$\begingroup$ @user125763: trivial residue field extension at the marked points (so ensures induced map on completions from the etale morphism is an isomorphism). $\endgroup$ Commented Jun 10, 2014 at 1:33