Title: Rationality of algebraic varieties over non-closed fields
Abstract: The most basic algebraic varieties are projective spaces, and their closest relatives are rational varieties. These are varieties that admit a 1-to-1 parametrization by projective space on a dense open subset; hence, 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), and the classical Lüroth problem was concerned with determining the rationality of certain varieties. Over fields that are not algebraically closed, 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 dimension 3, there are no such known criteria, even after restricting to threefolds that become rational over the algebraic closure of the ground field. In this talk, I will first give a survey of some results on rationality of algebraic varieties. Then I will explain results studying a rationality criteria for 3-dimensional varieties over non-closed fields.
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