3
$\begingroup$

Let $S$ be a surface (so a $2$-dimensional proper $k$-scheme) and $s$ a singular point which is a rational double point.

One common characterisation of a RDP is that under sufficient conditions there exist a resultion morphsim $r: \tilde{S}\to S$ of surfaces such that the fiber $r^{-1}(s)$ is an set consisting of rational curves with self-intersection-number $−2$. This leads to classification of such RDP's via Dynkin-diagrams for ADE-curves.

I'm keen interested in geometrical intuition behind the RDP's. Concretely, my question is if there exist characterisation for a rational double point on the level of it's stalk $\mathcal{O}_{S,s}$ which "reflects" in some way the geometry of this rational double point?

Namely, since $S$ is proper and therefore finitely generated $k$-algebra, can the stalk $\mathcal{O}_{S,s}$ (or maybe at least it's completion or base change to algebraically closed field) obtain the shape like $k[x,y,z]_s/(f)$ where $f$ represents the curve/geometry of this singularity? If yes, what kind of polynomial $f$ might be?

Why it is called a double point? Does this mean that $f$ is contained in in the square $m_s^2$ of the unique maximal ideal $m_s$ but not in $m_s^3$?

Or is it a too naive approach in order to understand intuitively the "nature" of RDP's?

Improve this question
$\endgroup$
6
  • $\begingroup$ The completion is indeed a hypersurface and they can be written down explicitly. For example, the $A_n$ singularities have equations $z^{n+1}=xy$. Similar equations exist for $D_n$s and the $E$ s. $\endgroup$
    – Mohan
    Commented May 11, 2019 at 0:10
  • $\begingroup$ @Mohan:so the "geometry" and meaning of "double" indeed only crystallize out after completion? following point: which conditions should $S$ fullfil to garantee that the completions of stalks $O_{S,s}^{\wedge} $ are from the shape $k[[x,y,z]]/(f)$? $\endgroup$
    – user267839
    Commented May 11, 2019 at 0:58
  • $\begingroup$ More concretely in light of this thread: mathoverflow.net/questions/191725/… I'm looking for conditions for $S$ such that $O_{S,s}^{\wedge}= k[[x,y,z]]/(f)$ where if $s$ is regular then $f$ is linear and therefore wlog $O_{S,s}^{\wedge}= k[[x,y,z]]/(f) \cong k[[x,y]]$. Otherwise $f \in m_s ^2 \backslash m_s^3$. $\endgroup$
    – user267839
    Commented May 11, 2019 at 0:59
  • 1
    $\begingroup$ I would suggest that you look at the classic paper of M. Artin on rational singularities. $\endgroup$
    – Mohan
    Commented May 11, 2019 at 2:01
  • 2
    $\begingroup$ I would also recommend Fifteen characterizations of rational double points and simple critical points, by Alan Durfee (L'enseignement Math. 25 (1979) 131-163). $\endgroup$
    – abx
    Commented May 11, 2019 at 4:40

0

Reset to default

You must log in to answer this question.