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SUMMARY:Motivic classes of curvilinear  Hilbert schemes - Ilaria Rossinell
 i (EPFL - Ecole Polytechnique Fédérale de Lausanne)
DTSTART:20240620T103000Z
DTEND:20240620T113000Z
UID:TALK217687@talks.cam.ac.uk
DESCRIPTION:The arc scheme X&infin\; of a singular variety (X\, 0) is char
 acterized by the fact that the set of K-points X&infin\;(K) is in bijectio
 n with the set Hom(Spec(K[[t]])\, X) of K[[t]]-points of the variety. Capt
 uring a lot of the geometric behavior of the singularity\, we work with th
 e 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).\nIn this talk\, we specif
 ically focus on the curvilinear Hilbert schemes of Hilbk0(X). We discuss t
 he construction of a geometric bijection relating truncated punctual\, smo
 oth arcs with curvilinear Hilbert schemes. This allows us to express certa
 in Igusa zeta functions in terms of series of motivic classes of the curvi
 linear component\, and vice versa obtain a recursive formula to compute mo
 tivic classes of curvilinear Hilbert schemes in terms of an embedded resol
 ution of singularities. We also mention some extensions to monomial Hilber
 t schemes (joint with G. B&eacute\;rczi) as well as to the full Hilbert sc
 heme.\nIn addition to this\, we quickly discuss curvilinear Hilbert scheme
 s 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.
LOCATION:Seminar Room 1\, Newton Institute
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