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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: 2014-05-302016-09-22 Date Deposited: 2014-05
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