| Résume | 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 the punctual Hilbert scheme of (X, 0).
In this talk, we specifically focus on the curvilinear Hilbert schemes of Hilb^k_0(X). We discuss the construction of a geometric bijection relating truncated punctual smooth arcs with curvilinear schemes, and punctual arcs with principal schemes. This allows us to express certain Igusa zeta functions in terms of a series of motivic classes of the curvilinear component and, vice versa, obtain a formula to compute motivic classes of curvilinear Hilbert schemes in terms of an embedded resolution of singularities of (X, 0).
In addition to this, we discuss curvilinear and principal Hilbert schemes in the context of plane curve singularities, where we examine the construction of a geometric bijection relating punctual arcs with principal schemes, and then obtain the same results for principal schemes and the classic Igusa zeta function. This integration technique is also employed to construct new topological polynomial invariants of curve singularities, that we try to interpret in view of
a conjecture proposed by Oblomkov, Rasmussen and Shende. |
