Geometry Over Non-Closed Fields: February 26-March 3, 2012

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.


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


Constructing rational curves

Cone of curve classes

Geometry of spaces of rational curves

Arithmetic applications