Arc spaces of cA-type singularities
Jennifer M. Johnson (Princeton Univ), J\'anos Koll\'ar (Princeton, Univ)
TL;DR
This paper investigates the structure of arc spaces on certain singularities defined by xy=f(z_1,...,z_n), revealing the number of irreducible components and conditions under which the Nash map is not surjective.
Contribution
It establishes a precise count of irreducible components of arc spaces and identifies conditions affecting the Nash map's surjectivity for cA-type singularities.
Findings
Number of irreducible components equals multiplicity of f minus 1.
Nash map is not surjective if n>1 and leading term of f is not a perfect square.
Provides new insights into the geometry of arc spaces on cA-type singularities.
Abstract
We study the space of arcs on a singularity of the form xy=f(z_1,..., z_n) and prove 2 main results. (i) The number of irreducible components equals the multiplicity of f minus 1. (ii) If n>1 and the leading homogeneous term of f is not a perfect square then the Nash map from the set of irreducible components to the set of essential divisors is not surjective.
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Taxonomy
TopicsAdvanced Topics in Algebra · Advanced Differential Equations and Dynamical Systems · Finite Group Theory Research