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.


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


Zbl 0719.14005
Full Text: DOI


[1] Eduardo Casas-Alvero, Singularities of plane curves, London Mathematical Society Lecture Note Series, vol. 276, Cambridge University Press, Cambridge, 2000. · Zbl 0967.14018
[2] Jesús Fernández-Sánchez, On sandwiched singularities and complete ideals, J. Pure Appl. Algebra 185 (2003), no. 1-3, 165 – 175. · Zbl 1066.14041
[3] J. FERNÁNDEZ-SÁNCHEZ, Nash families of smooth arcs on a sandwiched singularity. To appear in Math. Proc. Cambridge. Philos. Soc. · Zbl 1074.14031
[4] Shihoko Ishii and János Kollár, The Nash problem on arc families of singularities, Duke Math. J. 120 (2003), no. 3, 601 – 620. · Zbl 1052.14011
[5] John F. Nash Jr., Arc structure of singularities, Duke Math. J. 81 (1995), no. 1, 31 – 38 (1996). A celebration of John F. Nash, Jr. · Zbl 0880.14010
[6] C. PLÉNAT, A propos de la conjecture de Nash. Preprint. math.AG/0301358.
[7] Mark Spivakovsky, Sandwiched singularities and desingularization of surfaces by normalized Nash transformations, Ann. of Math. (2) 131 (1990), no. 3, 411 – 491. · Zbl 0719.14005
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