Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry the study of lines and conics. From the modern standpoint, these areas are synthesized in the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that the arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves and families of rational curves on it. One incarnation of this insight is Lang’s philosophy, which continues to drive modern research in this area: hyperbolic varieties have few rational points. Another is Grothendieck’s anabelian geometry: hyperbolic varieties are characterized by their ´etale fundamental groups, and rational points correspond to Galois-theoretic sections. The discussion of Geometry of nonclosed fields will focus on the intertwined manifestations of these aspects of higher-dimensional arithmetic geometry.

The focus of the first meeting is the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Topics include: rational connectedness and simply connectedness, rational curves on log-varieties, rationally connected quotients of spaces of rational curves, degenerations of spaces of rational curves, rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties. Possible applications include: existence of rational points over function fields of curves and surfaces, potential density of rational points over global fields, weak and strong approximation.

## Participants

Dan Abramovich | Brown University |

Fedor Bogomolov | New York University |

Jean-Louis Colliot-Thélène | Orsay |

Izzet Coskun | University of Illinois, Chicago |

Olivier Debarre | École Normale Supérieure, Paris |

Tom Graber | California Institute of Technology |

Brendan Hassett | Rice University |

Stefan Kebekus | Albert-Ludwigs-Universität Freiburg |

Sándor Kovács | University of Washington |

Jun Li | Stanford University |

Max Lieblich | University of Washington |

Christian Liedtke | University of Düsseldorf |

James McKernan | Massachuetts Institute of Technology |

Martin Olsson | University of California, Berkeley |

Jason Starr | State University of New York, Stony Brook |

Burt Totaro | University of Cambridge |

Yuri Tschinkel | New York University |

Ravi Vakil | Stanford University |

Anthony Varilly-Alvarado | Rice University |

Chenyang Xu | University of Utah |

## Agenda & Notes

### Foundations

**Dan Abramovich**

*Logarithmic stable maps*(PDF)**Chanyang Xu**

*Irreducibility and degenerate fibers of Fano fibrations*(PDF)**Burt Totaro**

*The integral Hodge conjecture for threefolds*(PDF)

### Constructing rational curves

**James McKernan**

*MMP and rational curves*(PDF)**Christian Liedtke**

*Constructing rational curves on K3 surfaces*(PDF)

### Cone of curve classes

**Izzet Coskun**

*MMP for the Hilbert scheme of points*(PDF)

### Geometry of spaces of rational curves

**Stefan Kebekus**

*Uniruledness criteria and applications to classification and foliations*(PDF)**Olivier Debarre**

*Curves of low degree on projective varieties*(PDF)

### Arithmetic applications

**Jean-Louis Colliot-Thélène**

*Brauer-Manin obstructions and integral points*(PDF)**Anthony Várilly-Alvarado**

*Transcendental obstructions on K3 surfaces*(PDF)**Max Lieblich**

*The period-index problem for Severi-Brauer varieties*(PDF)**Jason Starr**

*Rational points over function fields of curves and surfaces***Fedor Bogomolov**

*On the section conjecture*