Fernández-Sánchez, Jesús Equivalence of the Nash conjecture for primitive and sandwiched singularities. (English) Zbl 1056.14004 Proc. Am. Math. Soc. 133, No. 3, 677-679 (2005). 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. Cited in 6 Documents 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 Citations:Zbl 0719.14005 PDF BibTeX XML Cite \textit{J. Fernández-Sánchez}, Proc. Am. Math. Soc. 133, No. 3, 677--679 (2005; Zbl 1056.14004) Full Text: DOI References: [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.