On an analog of Pinkham's theorem for non-Tjurina components of rational singularities

Altınok, Selma

Comptes Rendus. Mathématique, Tome 347 (2009) no. 11-12, pp. 643-646.

Il existe une correspondance entre l'ensemble des fonctions de l'idéal maximal de l'anneau local en une singularité rationnelle ξ d'une surface et un ensemble ${\mathcal{E}}^{+}(E)$ de diviseurs effectifs portés par la fibre exceptionnelle E d'une résolution de cette singularité. Étant donné un élément $Y\in {\mathcal{E}}^{+}(E)$ et une composante N de Y qui n'est pas Tjurina, nous établissons une formule donnant le plus petit élément de l'ensemble des diviseurs $X\in {\mathcal{E}}^{+}(E)$ supérieur ou égal à $Y+N$, indiquée mais non démontrée dans Tosun (1999).

There is a correspondence between the set of functions in the maximal ideal of the local ring of a rational surface singularity ξ and the set ${\mathcal{E}}^{+}(E)$ consisting of certain effective divisors supported on the exceptional fiber E of a resolution of the singularity. Given an element $Y\in {\mathcal{E}}^{+}(E)$ and a non-Tjurina component N of Y, we verify a formula for the least element of the set of divisors $X\in {\mathcal{E}}^{+}(E)$ greater than or equal to $Y+N$ stated but not proved in Tosun (1999).

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