Deformations of the Hilbert scheme of points on a del Pezzo surface
Li, Chunyi
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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
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.
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