Originating from the School on Birational Geometry of Hypersurfaces, this volume focuses on the notion of (stable) rationality of projective varieties and, more specifically, hypersurfaces in projective spaces, and provides a large number of open questions, techniques and spectacular results.The aim of the school was to shed light on this vast area of research by concentrating on two main aspects: (1) Approaches focusing on (stable) rationality using deformation theory and Chow-theoretic tools like decomposition of the diagonal; (2) The connection between K3 surfaces, hyperkähler geometry and cubic fourfolds, which has both a Hodge-theoretic and a homological side.Featuring the beautiful lectures given at the school by Jean-Louis Colliot-Thélène, Daniel Huybrechts, Emanuele Macrì, and Claire Voisin, the volume also includes additional notes by János Kollár and an appendix by Andreas Hochenegger.
Foreword .- Part 1 Birational invariants and (stable) rationality .- 1. Claire Voisin : Birational invariants and decomposition of the diagonal.- 2. Jean-Louis Colliot-Thélène : Non rationalité stable sur le corps qualconques.- 3 Jean-Louis Colliot-Thélène : Introduction to work of Hassett-Pirutka-Tschinkel and Schreieder.- Part 2 Hypersurfaces .- 4. János Kollár : The rigidity theorem of Fano-Segre-Iskovskikh-Manin-Pukhlikov-Corti-Cheltsov-De Fernex-Ein-Mustata-Zhuang.- 5. Daniel Huybrechts : Hodge theory of cubic fourfolds, their Fano varieties, and associated K3 categories.- 6. Emanuele Macrì, Paolo Stellari: Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces.- Appendix: Andreas Hochenegger : Introduction to derived categories of coherent sheaves.
Foreword .- Part 1 Birational invariants and (stable) rationality .- 1. Claire Voisin : Birational invariants and decomposition of the diagonal.- 2. Jean-Louis Colliot-Thélène : Non rationalité stable sur le corps qualconques.- 3 Jean-Louis Colliot-Thélène : Introduction to work of Hassett-Pirutka-Tschinkel and Schreieder.- Part 2 Hypersurfaces .- 4. János Kollár : The rigidity theorem of Fano-Segre-Iskovskikh-Manin-Pukhlikov-Corti-Cheltsov-De Fernex-Ein-Mustata -Zhuang.- 5. Daniel Huybrechts : Hodge theory of cubic fourfolds, their Fano varieties, and associated K3 categories.- 6. Emanuele Macrì, Paolo Stellari: Lectures on non-commutative K3 surfaces, Bridgeland stability, and moduli spaces.- Appendix: Andreas Hochenegger : Introduction to derived categories of coherent sheaves.
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