## Equivalence of the Nash conjecture for primitive and sandwiched singularities.(English)Zbl 1056.14004

From the text: A normal surface singularity $$(X, Q)$$ is said to be sandwiched if it dominates birationally a non-singular surface. They arise when a complete $${\mathbf m}$$-primary ideal in a local regular $$\mathbb{C}$$-algebra $$R$$ of dimension two is blown up. A sandwiched singularity is said to be primitive if it can be obtained by blowing up a simple ideal, that is, a complete irreducible ideal of $$R$$. It is known that any sandwiched singularity is the birational join of some primitive singularities [M. Spivakovsky, Ann. Math. (2) 131, 411–491 (1990; Zbl 0719.14005)]. In this note, we prove that the Nash conjecture for sandwiched singularities and for primitive singularities are equivalent.

### MSC:

 14B05 Singularities in algebraic geometry 14E15 Global theory and resolution of singularities (algebro-geometric aspects) 14J17 Singularities of surfaces or higher-dimensional varieties

### Keywords:

arc; infinitely near point; birational join

Zbl 0719.14005
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### References:

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