{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/49684"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/49684","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Deformations of the Hilbert scheme of points on a 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.","abstract_html":"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<span class=\"etd-inline-math\"><sup>n</sup> S</span> 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<span class=\"etd-inline-math\"><sup>n</sup> S</span> carries a generically symplectic holomorphic Poisson structure. Moreover, the generic deformation of Hilb<span class=\"etd-inline-math\"><sup>n</sup>S</span> 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.","abstract_has_math":true,"creators":["Li, Chunyi"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Nevins, Thomas A.","Katz, Sheldon","Bradlow, Steven B.","Schenck, Henry K."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-05-30T17:04:43Z","date_published":"2014-05-30T17:04:43Z","updated_at":"2026-07-22T22:25:38Z","subjects":["Hilbert scheme","deformation theory","del Pezzo surface"],"languages":["en"],"rights":["Copyright 2014 Chunyi Li"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/49684","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Nevins, Thomas A.","Katz, Sheldon","Bradlow, Steven B.","Schenck, Henry K."]},{"key":"dc:creator","label":"Author","values":["Li, Chunyi"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-05-30T17:04:43Z","2016-09-22T20:59:22Z","2014-05"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Hilbert scheme","deformation theory","del Pezzo surface"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2014 Chunyi Li"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/49684"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2014-04-23T14:15:57Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Li_Chunyi.pdf: 405581 bytes, checksum: 10f052f247d1043296ec90877bd0f648 (MD5)","Made available in DSpace on 2014-05-30T17:04:43Z (GMT). No. of bitstreams: 2 Chunyi_Li.pdf: 405593 bytes, checksum: aa9392e4ae905312c5e3ea5c429d35cb (MD5) license.txt: 4056 bytes, checksum: 62f1b0fac12c0a836bedc221fe2ffdf6 (MD5)","Restriction data tranferred 2014-07-01T11:38:37-05:00 Original Data Group with Access UIUC Users [automated] Release Date: 2016-05-30 12:09:03 UTC Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Seth Robbins (robbins.sd@gmail.com) on 2014-05-30T17:09:34Z Item is restricted until 2016-05-30T17:09:03Z","U of I Only Restriction Lifted for Item 49735 on 2016-09-22T20:59:22Z."]},{"key":"dc:title","label":"Title","values":["Deformations of the Hilbert scheme of points on a del Pezzo surface"]}]}],"canonical_facts":{"dc:contributor":["Nevins, Thomas A.","Katz, Sheldon","Bradlow, Steven B.","Schenck, Henry K."],"dc:creator":["Li, Chunyi"],"dc:date":["2014-05-30T17:04:43Z","2016-09-22T20:59:22Z","2014-05"],"dc:description":["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.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2014-04-23T14:15:57Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Li_Chunyi.pdf: 405581 bytes, checksum: 10f052f247d1043296ec90877bd0f648 (MD5)","Made available in DSpace on 2014-05-30T17:04:43Z (GMT). No. of bitstreams: 2 Chunyi_Li.pdf: 405593 bytes, checksum: aa9392e4ae905312c5e3ea5c429d35cb (MD5) license.txt: 4056 bytes, checksum: 62f1b0fac12c0a836bedc221fe2ffdf6 (MD5)","Restriction data tranferred 2014-07-01T11:38:37-05:00 Original Data Group with Access UIUC Users [automated] Release Date: 2016-05-30 12:09:03 UTC Reason: Author requested U of Illinois access only (OA after 2yrs) in Vireo ETD system","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Seth Robbins (robbins.sd@gmail.com) on 2014-05-30T17:09:34Z Item is restricted until 2016-05-30T17:09:03Z","U of I Only Restriction Lifted for Item 49735 on 2016-09-22T20:59:22Z."],"dc:identifier":["http://hdl.handle.net/2142/49684"],"dc:language":["en"],"dc:rights":["Copyright 2014 Chunyi Li"],"dc:subject":["Hilbert scheme","deformation theory","del Pezzo surface"],"dc:title":["Deformations of the Hilbert scheme of points on a del Pezzo surface"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:38Z"}