Fano varieties are algebraic varieties with positive curvature; they are basic building blocks of algebraic varieties. Great progress has been recently made by Xu et al. to construct moduli spaces of Fano varieties by using K-stability (which is related to the existence of Kähler-Einstein metrics). These moduli spaces are called K-moduli. In this talk I will explain how to easily deduce some geometric properties of K-moduli by using toric geometry and deformation theory. In particular, I will show how to construct a 1-dimensional component of K-moduli which parametrises certain K-polystable del Pezzo surfaces. * ZOOM information will not be provided. Please send an email to Jinhyung Park if you are interested in.

In this talk, we will introduce the absolute coregularity of Fano varieties. The coregularity measures the singularities of the anti-pluricanonical sections. Philosophically, most Fano varieties have coregularity 0. In the talk, we will explain some theorems that support this philosophy. We will show that a Fano variety of coregularity 0 admits a non-trivial section in |-2K_X|, independently of the dimension of X. This is joint work with Fernando Figueroa, Stefano Filipazzo, and Junyao Peng. * ZOOM information will not be provided. Please send an email to Jinhyung Park if you are interested in.

The classification of terminal Fano 3-folds has been tackled from different directions: for instance, using the Minimal Model Program, via explicit Birational Geometry, and via Graded Rings methods. In this talk I would like to introduce the Graded Ring Database - an upper bound to the numerics of Fano 3-folds - and discuss the role it plays in the classification and construction of codimension 4 Fano 3-folds having Fano index 2.

Castelnuovo-Mumford regularity, simply regularity, is one of the most interesting invariants in projective algebraic geometry, and the regularity conjecture due to Eisenbud and Goto says that the regularity can be controlled by the degree for any projective variety. But counterexamples to the conjecture have been constructed by some methods. In this talk we review the counterexample constructions including the Rees-like algebra method by McCullough and Peeva and the unprojection method.