Moduli spaces of K3 surfaces and cubic fourfolds
Moduli spaces of K3 surfaces and cubic fourfolds
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DissertationDate of Exam
16.10.2019Date of Publication
14.11.2019Advisor
Huybrechts, DanielCo-Referee
Oberdieck, GeorgMetadata
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Abstract
This thesis is concerned with the Hodge-theoretic relation between polarized K3 surfaces of degree d and special cubic fourfolds of discriminant d, as introduced by Hassett.
For half of the d, K3 surfaces associated to cubic fourfolds come naturally in pairs. As our first main result, we prove that if (S,L) and (St,Lt) form such a pair of polarized K3 surfaces, then St is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3,L,d/6). We also explain for which d the Hilbert schemes Hilbn(S) and Hilbn(St) are birational.
Next, we study the more general concept of associated twisted K3 surfaces. Our main contribution here is the construction of moduli spaces of polarized twisted K3 surfaces of fixed degree and order. We strengthen a theorem of Huybrechts about the existence of associated twisted K3 surfaces. We show that like in the untwisted situation, half of the time, associated twisted K3 surfaces come in pairs, and we explain how the elements of such a pair are related to each other.
For half of the d, K3 surfaces associated to cubic fourfolds come naturally in pairs. As our first main result, we prove that if (S,L) and (St,Lt) form such a pair of polarized K3 surfaces, then St is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3,L,d/6). We also explain for which d the Hilbert schemes Hilbn(S) and Hilbn(St) are birational.
Next, we study the more general concept of associated twisted K3 surfaces. Our main contribution here is the construction of moduli spaces of polarized twisted K3 surfaces of fixed degree and order. We strengthen a theorem of Huybrechts about the existence of associated twisted K3 surfaces. We show that like in the untwisted situation, half of the time, associated twisted K3 surfaces come in pairs, and we explain how the elements of such a pair are related to each other.
Subjects
Algebraische Geometrie, K3-Flächen, Kubische 4-Mannigfaltigkeiten, Brauergruppen, Modulräume, Algebraic geometry, K3 surfaces, cubic fourfolds, Brauer groups, Moduli spaces
Classification (DDC)
510 MathematikBrakkee, Emma: Moduli spaces of K3 surfaces and cubic fourfolds. - Bonn, 2019. - Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn.
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-56400
Online-Ausgabe in bonndoc: https://nbn-resolving.org/urn:nbn:de:hbz:5n-56400
@phdthesis{handle:20.500.11811/8103,
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-56400,
author = {{Emma Brakkee}},
title = {Moduli spaces of K3 surfaces and cubic fourfolds},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2019,
month = nov,
note = {This thesis is concerned with the Hodge-theoretic relation between polarized K3 surfaces of degree d and special cubic fourfolds of discriminant d, as introduced by Hassett.
For half of the d, K3 surfaces associated to cubic fourfolds come naturally in pairs. As our first main result, we prove that if (S,L) and (St,Lt) form such a pair of polarized K3 surfaces, then St is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3,L,d/6). We also explain for which d the Hilbert schemes Hilbn(S) and Hilbn(St) are birational.
Next, we study the more general concept of associated twisted K3 surfaces. Our main contribution here is the construction of moduli spaces of polarized twisted K3 surfaces of fixed degree and order. We strengthen a theorem of Huybrechts about the existence of associated twisted K3 surfaces. We show that like in the untwisted situation, half of the time, associated twisted K3 surfaces come in pairs, and we explain how the elements of such a pair are related to each other.},
url = {https://hdl.handle.net/20.500.11811/8103}
}
urn: https://nbn-resolving.org/urn:nbn:de:hbz:5n-56400,
author = {{Emma Brakkee}},
title = {Moduli spaces of K3 surfaces and cubic fourfolds},
school = {Rheinische Friedrich-Wilhelms-Universität Bonn},
year = 2019,
month = nov,
note = {This thesis is concerned with the Hodge-theoretic relation between polarized K3 surfaces of degree d and special cubic fourfolds of discriminant d, as introduced by Hassett.
For half of the d, K3 surfaces associated to cubic fourfolds come naturally in pairs. As our first main result, we prove that if (S,L) and (St,Lt) form such a pair of polarized K3 surfaces, then St is isomorphic to the moduli space of stable coherent sheaves on S with Mukai vector (3,L,d/6). We also explain for which d the Hilbert schemes Hilbn(S) and Hilbn(St) are birational.
Next, we study the more general concept of associated twisted K3 surfaces. Our main contribution here is the construction of moduli spaces of polarized twisted K3 surfaces of fixed degree and order. We strengthen a theorem of Huybrechts about the existence of associated twisted K3 surfaces. We show that like in the untwisted situation, half of the time, associated twisted K3 surfaces come in pairs, and we explain how the elements of such a pair are related to each other.},
url = {https://hdl.handle.net/20.500.11811/8103}
}
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