Exceptional Collections and Rationality

Explore the intriguing relationship between exceptional collections and rationality in algebraic geometry through this 52-minute lecture by Matthew Ballard from the University of South Carolina. Delve into Orlov's expectation that smooth projective varieties with full exceptional collections must be rational over their base field, and examine this concept in the context of non-closed fields. Discover the existence of smooth projective geometrically rational 3-folds with full etale-exceptional collections but no points over the base field. Learn how the expectation holds true for smooth projective toric varieties over any base field, demonstrating that k-rationality is implied by the presence of a full k-exceptional collection. Gain insights into this collaborative research conducted with A. Duncan, A. Lamarche, and P. McFaddin, presented at the University of Miami on January 30, 2020.