University of Illinois Urbana-Champaign

Deformations of the Hilbert scheme of points on a del Pezzo surface

Li, Chunyi

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Chunyi_Li.pdf

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https://hdl.handle.net/2142/49684

Description

Title
Deformations of the Hilbert scheme of points on a del Pezzo surface
Author(s)
Li, Chunyi
Issue Date
2014-05-30T17:04:43Z
Director of Research (if dissertation) or Advisor (if thesis)
Nevins, Thomas A.
Doctoral Committee Chair(s)
Katz, Sheldon
Committee Member(s)
  • Nevins, Thomas A.
  • Bradlow, Steven B.
  • Schenck, Henry K.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Date of Ingest
2014-05-30T17:04:43Z
Keyword(s)
  • Hilbert scheme
  • deformation theory
  • del Pezzo surface
Abstract
The Hilbert scheme of $n$ points in a smooth del Pezzo surface $S$ parameterizes zero-dimensional subschemes with length $n$ on $S$. We construct a flat family of deformations of Hilb$^n S$ which can be conceptually understood as the family of Hilbert schemes of points on a family of noncommutative deformations of $S$. Further we show that each deformed Hilb$^n S$ carries a generically symplectic holomorphic Poisson structure. Moreover, the generic deformation of Hilb$^nS$ has a $(k+2)$-dimensional moduli space, where the del Pezzo surface is the blow up of projective plane at $k$ sufficiently general points; and each of the fibers is of the form that we construct. Our work generalizes results of Nevins-Stafford constructing deformations of the Hilbert scheme of points on the plane, and of Hitchin studying those deformations from the viewpoint of Poisson geometry.
Graduation Semester
2014-05
Permalink
http://hdl.handle.net/2142/49684
Copyright and License Information
Copyright 2014 Chunyi Li

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