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A support theorem for Hilbert schemes of planar curves, II

Part of: Projective and enumerative geometry Singularities Cycles and subschemes Curves

Published online by Cambridge University Press:  28 April 2021

Luca Migliorini ,
Vivek Shende  and
Filippo Viviani
Luca Migliorini
Affiliation:
Department of Mathematics, Università di Bologna, 40126Bologna, [email protected]
Vivek Shende
Affiliation:
Department of Mathematics, University of California, Berkeley, CA94720, [email protected]
Filippo Viviani
Affiliation:
Department of Mathematics, Università di Roma Tre, 00146Roma, [email protected]
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Abstract

We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.

Keywords

compactified JacobiansHilbert scheme of pointsreduced curves with locally planar singularitiesperverse filtrationdecomposition theoremsupport theorem

MSC classification

Primary: 14H40: Jacobians, Prym varieties 14C05: Parametrization (Chow and Hilbert schemes) 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 32S60: Stratifications; constructible sheaves; intersection cohomology
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 4 , April 2021 , pp. 835 - 882
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Copyright
© The Author(s) 2021

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Footnotes

L.M. is partially supported by PRIN project 2015 ‘Spazi di moduli e teoria di Lie’. During the (long) preparation of this paper L.M. was a member of the School of Mathematics of the Institute for Advanced Study in Princeton, partially funded by the Giorgio and Elena Petronio fellowship. V. S. is supported by the NSF grant DMS-1406871, and by a Sloan fellowship. F.V. is partially supported by PRIN ‘Geometria delle varietà algebriche’ and GNSAGA-INdAM.

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