Dokument: The Kummer Constructions in Families
| Titel: | The Kummer Constructions in Families | |||||||
| URL für Lesezeichen: | https://docserv.uni-duesseldorf.de/servlets/DocumentServlet?id=63084 | |||||||
| URN (NBN): | urn:nbn:de:hbz:061-20230728-081344-3 | |||||||
| Kollektion: | Dissertationen | |||||||
| Sprache: | Englisch | |||||||
| Dokumententyp: | Wissenschaftliche Abschlussarbeiten » Dissertation | |||||||
| Medientyp: | Text | |||||||
| Autor: | Bergqvist, Jakob Studsgaard [Autor] | |||||||
| Dateien: |
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| Beitragende: | Prof. Dr. Stefan Schröer [Gutachter] Prof. Dr. Markus Reineke [Gutachter] | |||||||
| Stichwörter: | 14-xx Algebraic geometry, 14J10 Families, moduli, classification: algebraic theory, 14J28 $K3$ surfaces and Enriques surfaces, 14J17 Singularities, 14L30 Group actions on varieties or schemes (quotients) | |||||||
| Dewey Dezimal-Klassifikation: | 500 Naturwissenschaften und Mathematik » 510 Mathematik | |||||||
| Beschreibung: | The aim of this thesis is to study generalized Kummer constructions in families. This involves many different overlapping areas of study and I introduce them in turn. I provide a thorough introduction to the theory of group schemes, focusing in particular on finite and diagonalizable group schemes. Then, the focus turns to group scheme actions and quotients by these with the existence of quotients being discussed. I explain how an action by a diagonalizable group scheme may be viewed as a grading, and how one may use this to compute quotients. It is also outlined how one may interpret group scheme actions using Lie algebras and how this technique allows for computing a quotient. I briefly outline some general surface theory, focusing on rational double point singularities, and the conditions $(S_i)$ and $(R_i)$ of Serre. To give perspective, the thesis includes a detailed explanation of the classical Kummer construction over fields, and later an outline of how the classical construction behaves in families. Turning the focus to families, I study quotients of families. Special attention is paid to when taking fibers and quotients commute. I discuss simultaneous resolution of singularities, as a preparation for the main result of the thesis. I outline the generalized Kummer constructions in characteristic $2$ with $\alpha_2$ and $\mu_2$ of Schröer and Schröer and Kondo. The thesis then concludes with the new results; I show that the family of singular surfaces $(C\times C)/\mu_2$, with $C$ the cuspidal curve, admits a simultaneous resolution after a finite base change of degree at most $5184$, and conclude that the generalized Kummer construction with $\mu_2$ works in families after this base change. | |||||||
| Lizenz: |  Dieses Werk ist lizenziert unter einer Creative Commons Namensnennung 4.0 International Lizenz | |||||||
| Fachbereich / Einrichtung: | Mathematisch- Naturwissenschaftliche Fakultät » WE Mathematik » Algebraische Geometrie | |||||||
| Dokument erstellt am: | 28.07.2023 | |||||||
| Dateien geändert am: | 28.07.2023 | |||||||
| Promotionsantrag am: | 05.04.2023 | |||||||
| Datum der Promotion: | 19.06.2023 |
