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

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 Hilbⁿ 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 Hilbⁿ S carries a generically symplectic holomorphic Poisson structure. Moreover, the generic deformation of HilbⁿS 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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Li, Chunyi

Contributors dc:contributor

• Nevins, Thomas A.
• Katz, Sheldon
• Bradlow, Steven B.
• Schenck, Henry K.

• Hilbert scheme
• deformation theory
• del Pezzo surface

Statement dc:rights

• Copyright 2014 Chunyi Li

Language dc:language

en

Handle dc:identifier

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

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/49684