The Nash problem of arcs and the rational double points <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math>

Abstract

This paper deals with the Nash problem, which consists in comparing the number of families of arcs on a singular germ of surface <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>U</mml:mi> </mml:math> with the number of essential components of the exceptional divisor in the minimal resolution of this singularity. We prove their equality in the case of the rational double points <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>D</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> ).