## Geometry Over Nonclosed FieldsBased on the Simons Symposia held in 2015, the proceedings in this volume focus on rational curves on higher-dimensional algebraic varieties and applications of the theory of curves to arithmetic problems. There has been significant progress in this field with major new results, which have given new impetus to the study of rational curves and spaces of rational curves on K3 surfaces and their higher-dimensional generalizations. One main recent insight the book covers is the idea that the geometry of rational curves is tightly coupled to properties of derived categories of sheaves on K3 surfaces. The implementation of this idea led to proofs of long-standing conjectures concerning birational properties of holomorphic symplectic varieties, which in turn should yield new theorems in arithmetic. This proceedings volume covers these new insights in detail. |

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### Contents

1 | |

Lines on Cubic Hypersurfaces Over Finite Fields | 18 |

Perverse Sheaves of Categories and Nonrationality | 53 |

Divisor Classes and the Virtual Canonical Bundle for Genus 0 Maps | 97 |

A Stronger Derived Torelli Theorem for K3 Surfaces | 127 |

Morphisms to BrauerSeveri Varieties with Applications to Del Pezzo Surfaces | 157 |

Arithmetic of K3 Surfaces | 197 |

OddDimensional Cohomology with Finite Coefficients and Roots of Unity | 249 |

### Other editions - View all

Geometry Over Nonclosed Fields Fedor Bogomolov,Brendan Hassett,Yuri Tschinkel No preview available - 2017 |

Geometry Over Nonclosed Fields Fedor Bogomolov,Brendan Hassett,Yuri Tschinkel No preview available - 2019 |

### Common terms and phrases

1-morphism algebraic Am(X ample cone automorphism Br(k Brauer group Brauer–Severi surface Brauer–Severi variety bundle canonical Cartier divisor characteristic complex K3 surface components compute contains Corollary corresponding cubic fourfold cubic hypersurface cubic threefold defined over F deformations del Pezzo surface denote dimension divisor class eigenvalues elliptic curve elliptic fibration embedding étale cohomology exact sequence example exists fiber filtration finite field functor Galois global sections Hasse principle homomorphism hyperkähler manifold induced integer intersection invertible sheaf irreducible isomorphism k-rational k-rational point K3 surface Kobayashi Lagrangian fibration lattice Lemma LG model Math metric moduli space multiplication number field obtain pair perverse sheaf Pezzo surface Picard number Picº polynomial prime projective Proof Proposition pullback quadric quotient rank rational resp Sect sheaf of categories sheaves singular smooth cubic Spec strongly filtered surface of degree surjective Tate conjecture Theorem torsion trivial