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Title:Deformations of the Hilbert scheme of points on a del Pezzo surface
Author(s):Li, Chunyi
Director of Research:Nevins, Thomas A.
Doctoral Committee Chair(s):Katz, Sheldon
Doctoral Committee Member(s):Nevins, Thomas A.; Bradlow, Steven B.; Schenck, Henry K.
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(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.
Issue Date:2014-05-30
URI:http://hdl.handle.net/2142/49684
Rights Information:Copyright 2014 Chunyi Li
Date Available in IDEALS:2014-05-30
2016-09-22
Date Deposited:2014-05


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