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 the punctual Hilbert scheme of (X, 0). This talk focuses on the curvilinear and principal Hilbert schemes of Hilbk 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 these components of the Hilbert scheme and, vice versa, obtain a formula to compute motivic classes of 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. This integration technique is 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.

Eventi passati

È possibile consultare gli eventi del precedente ciclo alla pagina dedicata