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)

Lecture 1. Boundedness in general type MMP and local K-stability

In this talk, I will discuss how to use local volumes from the local theory of K-stability to prove several boundedeness results in the Minimal Model Program for general type varieties. This is based on joint work with Jingjun Han and Ziquan Zhuang.

Lecture 2. Stable degeneration of Fano fibration germs

Fano fibrations can be viewed as an interpolation between Fano varieties and klt singularities, for which extremal contractions in the MMP provide more examples. In this talk, I will discuss the stable degeneration for Fano fibration germs. This provides a unified point of view for the degeneration theory in global and local K-stability, and verifies a conjecture of Sun and Zhang. This is based on joint work with Jiyuan Han, Minghao Miao, Linsheng Wang, and Tong Zhang.

• Lorenzo Barban (IBS-CCG) 

Chow quotients of flag varieties

The Chow quotient of a smooth projective variety under a torus action was introduced by Kapranov, Sturmfels and Zelevinsky with the aim to construct an intrinsic notion of quotient, which would not depend on a particular choice of a linearization as in the case of the Mumford’s GIT quotient. In this talk we report on a joint work with L. E. Solá Conde and G. Occhetta, where we explicitly compute the Chow quotient of the complete flag variety of vector subspaces of the 4-dimensional affine space under the action of the maximal torus of PGL(4). We also discuss its birational geometry, and the relation with the Chow quotients of partial flag varieties. 

• DongSeon Hwang (IBS-CCG)

Cascades on klt del Pezzo surfaces of Picard number one 

There have been numerous attempts to classify log del Pezzo surfaces. In this talk, I will quickly summarize these developments and present my work on the classification of klt del Pezzo surfaces of Picard number one. The result is obtained by generalizing the notion of ‘cascades’ of nonsingular del Pezzo surfaces, following the approach initiated by Miyanishi and Zhang. 

• Donghyeon Kim (Yonsei University)

Structure of the anti-canonical MMP of a pklt pair

We discuss the structure of the anti-canonical minimal model program. First, using the perturbation property of pklt triples, which follows from the existence of a quasi-monomial valuation computing the potential log discrepancy, we prove that the geometric generic fiber is pklt whenever the fibers are of \varepsilon-lc Calabi–Yau type over a Zariski-dense subset of the base. Surprisingly, although the geometric generic fiber is pklt, it need not be of klt Calabi–Yau type. Second, we investigate the structure of the anti-canonical MMP for a pklt surface and its minimal resolution.

• Haesong Seo (KAIST)

Hyperbolicity problems for homogeneous varieties with Picard number one

Kobayashi's conjecture predicts that a general hypersurface of sufficiently large degree in $\PP^n$ is hyperbolic, i.e., its Kobayashi pseudometric is nondegenerate. Meanwhile, by the monotonicity of the Kobayashi pseudometric under holomorphic maps, any curve of fixed genus in a hyperbolic projective manifold $X$ has a bounded degree. This motivates a more manageable hyperbolicity notion -- algebraic hyperbolicity. Accordingly, algebraic hyperbolicity has become a natural testing ground for Kobayashi's conjecture. For example, Voisin showed that a very general hypersurface in $\PP^n$ is algebraically hyperbolic when $d \geq 2n-1$ for $n \geq 4$. In this talk, we extend this program from $X = \PP^n$ to homogeneous varieties $X = G/P$ with Picard number one, and prove that hypersurfaces of the expected optimal degree are algebraically hyperbolic. This is based on joint work with Minseong Kwon.