Lena Ji, University of Michigan
"Rationality of algebraic varieties over non-algebraically-closed fields"
The most basic algebraic varieties are projective spaces, and their closest relatives are rational varieties. These are varieties that agree with affine space on a dense open subset, and hence have a coordinate system on this open subset. Thus, rational varieties are the easiest varieties to understand. Historically, rationality problems have been of great importance in algebraic geometry: for example, Severi was interested in finding rational parametrizations for moduli spaces of Riemann surfaces (algebraic curves). Over the complex numbers, techniques from geometry and topology can be used to extract invariants useful for rationality questions. Over fields that are not algebraically closed (such as the rational numbers), the arithmetic of the field adds additional subtleties to the rationality problem. When the dimension of the variety is at most 2, there are effective criteria to determine rationality. However, in higher dimensions, there are no such known criteria. In this talk, I will first give a survey of some results on rationality of algebraic varieties. Then I will explain results on rationality obstructions for higher-dimensional varieties that involve the arithmetic of the field.
M3 3127