{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71199"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71199","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Projections of Varieties","abstract":"This thesis is devoted to the study of varieties with the following property: if X is a smooth projectively normal variety in ('n), we say that X has the length one projection property if every isomorphic projection of X to ('n-1) has the property that the linear system cut out on it by the hypersurfaces of degree k is complete for k (GREATERTHEQ) 2.","abstract_html":"This thesis is devoted to the study of varieties with the following property: if X is a smooth projectively normal variety in ('n), we say that X has the length one projection property if every isomorphic projection of X to ('n-1) has the property that the linear system cut out on it by the hypersurfaces of degree k is complete for k (GREATERTHEQ) 2.","abstract_has_math":false,"creators":["Meadows, Catherine Ann"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-16T06:18:02Z","date_published":"2014-12-16T06:18:02Z","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8203532"],"render_values":[{"text":"(UMI)AAI8203532","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71199","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Meadows, Catherine Ann"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-16T06:18:02Z","10000-01-01","1981"]},{"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":["Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71199","(UMI)AAI8203532"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["This thesis is devoted to the study of varieties with the following property: if X is a smooth projectively normal variety in ('n), we say that X has the length one projection property if every isomorphic projection of X to ('n-1) has the property that the linear system cut out on it by the hypersurfaces of degree k is complete for k (GREATERTHEQ) 2.","The main result of this thesis is that, if X is a smooth variety in ('n), then the d-uple embedding of X has the length one projection property for large enough d. The theorem is first proved for any d-uple embedding of ('n) by induction on r, and is then extended to the d-uple embedding of any smooth variety for large enough d.","A theorem describing varieties with the length one projection property is also given. Suppose that X has the length one projection property non-trivially (i.e., that isomorphic projections of X do exist). Then for every variety Y such that X (L-HOOK) Y (L-HOOK) V(J), where J is the ideal generated by the quadratics in the defining ideal of X, we have Sec*(X) = Sec*(Y), where Sec*(Y) is defined to be the union of the secant lines through Y and the linear spaces tangent to Y. Thus in most cases we would expect the defining ideal of X to be minimal over its quadratic generators. An example is provided to show that this is not true of all cases.","Made available in DSpace on 2014-12-16T06:18:02Z (GMT). No. of bitstreams: 1 8203532.pdf: 3663506 bytes, checksum: f0beacbec1d89d4ce049e409b3c5ea6f (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 71365 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","149 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."]},{"key":"dc:title","label":"Title","values":["Projections of Varieties"]}]}],"canonical_facts":{"dc:creator":["Meadows, Catherine Ann"],"dc:date":["2014-12-16T06:18:02Z","10000-01-01","1981"],"dc:description":["This thesis is devoted to the study of varieties with the following property: if X is a smooth projectively normal variety in ('n), we say that X has the length one projection property if every isomorphic projection of X to ('n-1) has the property that the linear system cut out on it by the hypersurfaces of degree k is complete for k (GREATERTHEQ) 2.","The main result of this thesis is that, if X is a smooth variety in ('n), then the d-uple embedding of X has the length one projection property for large enough d. The theorem is first proved for any d-uple embedding of ('n) by induction on r, and is then extended to the d-uple embedding of any smooth variety for large enough d.","A theorem describing varieties with the length one projection property is also given. Suppose that X has the length one projection property non-trivially (i.e., that isomorphic projections of X do exist). Then for every variety Y such that X (L-HOOK) Y (L-HOOK) V(J), where J is the ideal generated by the quadratics in the defining ideal of X, we have Sec*(X) = Sec*(Y), where Sec*(Y) is defined to be the union of the secant lines through Y and the linear spaces tangent to Y. Thus in most cases we would expect the defining ideal of X to be minimal over its quadratic generators. An example is provided to show that this is not true of all cases.","Made available in DSpace on 2014-12-16T06:18:02Z (GMT). No. of bitstreams: 1 8203532.pdf: 3663506 bytes, checksum: f0beacbec1d89d4ce049e409b3c5ea6f (MD5) Previous issue date: 1981","Embargo set by: Seth Robbins for item 71365 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","149 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1981."],"dc:identifier":["http://hdl.handle.net/2142/71199","(UMI)AAI8203532"],"dc:subject":["Mathematics"],"dc:title":["Projections of Varieties"],"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":"2026-07-22T22:26:04Z"}