{"id":{"repo_id":"bielefeld","oai_identifier":"oai:pub.uni-bielefeld.de:2994518"},"canonical_url":"https://search.dev.ndltd.org/etd/bielefeld/oai:pub.uni-bielefeld.de:2994518","repository":{"repo_id":"bielefeld","name":"Universität Bielefeld","base_url":"https://pub.uni-bielefeld.de/oai"},"display":{"title":"Phantoms and Exceptional Collections on Rational Surfaces","abstract":"This thesis is concerned with exceptional collections on smooth projective surfaces. Any rational surface over an algebraically closed field admits a full exceptional collection. We extend certain classification results regarding exceptional collections, previously known for del Pezzo surfaces, to the blow-up of the projective plane in 9 very general points. More generally, we obtain a classification result for numerically exceptional collections of maximal length on smooth projective surfaces $X$ with $\\chi(\\mathcal{O}_X)=1$ and $K_X^2 + \\mathrm{rk}(\\mathsf{K}_0^\\mathrm{num}(X)) = 12$. In contrast to the case of 9 points, on the blow-up in 10 general points we construct an exceptional collection of maximal length which is not full. As a consequence, the orthogonal complement of this collection is a universal phantom category. This disproves a conjecture of Kuznetsov and a conjecture of Orlov.","abstract_html":"This thesis is concerned with exceptional collections on smooth projective surfaces. Any rational surface over an algebraically closed field admits a full exceptional collection. We extend certain classification results regarding exceptional collections, previously known for del Pezzo surfaces, to the blow-up of the projective plane in 9 very general points. More generally, we obtain a classification result for numerically exceptional collections of maximal length on smooth projective surfaces $X$ with <span class=\"etd-inline-math\">\\chi(\\mathcal{O}<sub>X</sub>)=1</span> and <span class=\"etd-inline-math\">K<sub>X</sub><sup>2</sup> + <span class=\"etd-inline-math-roman\">rk</span>(\\mathsf{K}<sub>0</sub><sup>\\</sup>mathrm{num}(X)) = 12</span>. In contrast to the case of 9 points, on the blow-up in 10 general points we construct an exceptional collection of maximal length which is not full. As a consequence, the orthogonal complement of this collection is a universal phantom category. This disproves a conjecture of Kuznetsov and a conjecture of Orlov.","abstract_has_math":true,"creators":["Krah, Johannes"],"institution":"Universität Bielefeld","degree_name":null,"degree_level":"thesis.doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2024,"date_issued":"2024-11-20","date_published":"2024-11-20","updated_at":"2026-07-27T18:50:01Z","subjects":[],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://pub.uni-bielefeld.de/record/2994518","outbound_label":"Repository record","outbound_source":"source_url"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Krah, Johannes"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["Universitätsbibliothek Bielefeld"]},{"key":"dc:type","label":"Dc Type","values":["doctoralThesis"]},{"key":"thesis:degree_level","label":"Degree Level","values":["thesis.doctoral"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Universität Bielefeld"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["This thesis is concerned with exceptional collections on smooth projective surfaces. Any rational surface over an algebraically closed field admits a full exceptional collection. We extend certain classification results regarding exceptional collections, previously known for del Pezzo surfaces, to the blow-up of the projective plane in 9 very general points. More generally, we obtain a classification result for numerically exceptional collections of maximal length on smooth projective surfaces $X$ with $\\chi(\\mathcal{O}_X)=1$ and $K_X^2 + \\mathrm{rk}(\\mathsf{K}_0^\\mathrm{num}(X)) = 12$. In contrast to the case of 9 points, on the blow-up in 10 general points we construct an exceptional collection of maximal length which is not full. As a consequence, the orthogonal complement of this collection is a universal phantom category. This disproves a conjecture of Kuznetsov and a conjecture of Orlov."]},{"key":"dc:format.medium","label":"Dc Format Medium","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Phantoms and Exceptional Collections on Rational Surfaces"]}]}],"canonical_facts":{"dc:creator":["Krah, Johannes"],"dc:description.abstract":["This thesis is concerned with exceptional collections on smooth projective surfaces. Any rational surface over an algebraically closed field admits a full exceptional collection. We extend certain classification results regarding exceptional collections, previously known for del Pezzo surfaces, to the blow-up of the projective plane in 9 very general points. More generally, we obtain a classification result for numerically exceptional collections of maximal length on smooth projective surfaces $X$ with $\\chi(\\mathcal{O}_X)=1$ and $K_X^2 + \\mathrm{rk}(\\mathsf{K}_0^\\mathrm{num}(X)) = 12$. In contrast to the case of 9 points, on the blow-up in 10 general points we construct an exceptional collection of maximal length which is not full. As a consequence, the orthogonal complement of this collection is a universal phantom category. This disproves a conjecture of Kuznetsov and a conjecture of Orlov."],"dc:format.medium":["application/pdf"],"dc:publisher":["Universitätsbibliothek Bielefeld"],"dc:title":["Phantoms and Exceptional Collections on Rational Surfaces"],"dc:type":["doctoralThesis"],"thesis:degree_level":["thesis.doctoral"],"thesis:institution_name":["Universität Bielefeld"]},"updated_at":"2026-07-27T18:50:01Z"}