Lecture series on Birational Geometry
2026. 8. 3-5
Yonsei University, Science Hall 225

• Joaquín Moraga (UCLA)
  
  

Lecture 1 . Fano Varieties and Coregularity
This lecture introduces Fano varieties and their role in the Minimal Model Program. After reviewing basic examples and fundamental invariants, we introduce the notion of coregularity, a new numerical invariant measuring the birational complexity of a Fano variety. We discuss its geometric interpretation, fundamental properties, and compute it in several classical examples, motivating the questions explored throughout the course.

Lecture 2. Measuring Birational Complexity

The second lecture develops the relationship between coregularity and the birational geometry of Fano varieties. We present recent results showing how coregularity reflects the complexity of birational models, complements, and log Calabi–Yau compactifications. Several examples illustrate how this invariant distinguishes different birational behaviors and provides a new framework for studying rationality questions and birational rigidity.

Lecture 3. Cluster Type Fano Varieties

The final lecture introduces cluster type Fano varieties, a class of Fano varieties closely related to cluster varieties and algebraic tori. We explain their geometric characterization, describe their birational properties, and discuss recent classification results. The lecture concludes with open problems and future directions connecting Fano varieties, cluster geometry, and mirror symmetry.

• Lu Qi (East China Normal University)
  
  

• Lorenzo Barban (IBS-CCG)
  
  

• DongSeon Hwang (IBS-CCG)
  
  

• Donghyeon Kim (Yonsei University)
  
  

• Haesong Seo (KAIST)