Abstract

The arc scheme X∞ of a singular variety (X, 0) is characterized by the fact that the set of K-points X∞(K) is in bijection with the set Hom(Spec(K[[t]]), X) of K[[t]]-points of the variety. Capturing a lot of the geometric behavior of the singularity, we work with the motivic measure on the arc scheme and Igusa zeta functions as we hope to provide a framework to unify the geometry of singular varieties with the geometry of punctual Hilbert schemes of (X, 0). In this talk, we specifically focus on the curvilinear Hilbert schemes of Hilbk0(X). We discuss the construction of a geometric bijection relating truncated punctual, smooth arcs with curvilinear Hilbert schemes. This allows us to express certain Igusa zeta functions in terms of series of motivic classes of the curvilinear component, and vice versa obtain a recursive formula to compute motivic classes of curvilinear Hilbert schemes in terms of an embedded resolution of singularities. We also mention some extensions to monomial Hilbert schemes (joint with G. Bérczi) as well as to the full Hilbert scheme. In addition to this, we quickly discuss curvilinear Hilbert schemes in the context of plane curve singularities. This integration technique is employed to construct new topological invariants of curve singularities, that we try to interpret in view of a conjecture proposed by Oblomkov, Rasmussen and Shende.