The 4-nodal cubic surface and certain surfaces of general type with geometric genus zero
After briefly describing the geometry of the 4-nodal cubic surface, I will introduce the Keum-Naie-Mendes Lopes-Pardini surfaces. Then I will talk about the bicanonical map, the deformation and moduli of these surfaces.
Introduction to Manin’s conjecture
Manin’s conjecture predicts the asymptotic distribution on the rational points on Fano varieties. In this talk, I will try to explain the historical development of the conjecture and explain the connection between geometry and arithmetic in the case of a log del Pezzo surface with A_4 and K_5 singularities. This is based on joint work in progress with Ulrich Derenthal.
K-stability of log del Pezzo surfaces with small alpha-invariants
In this talk, we estimate beta-invariants and delta-invariant of some singular log del Pezzo surfaces with quotient singularities. As a result we prove their K-stability and the existence of K\”ahler-Einstein metrics.
Fibrations of hyperelliptic curves of genus $3$ on minimal surfaces of general type with $p_g=0$
In this talk we consider results and examples of fibrations of hyperelliptic curves of genus $3$ with double fibers on minimal surfaces of general type with $p_g=0$. As its application we construct smooth minimal $3$-folds of general type canonically fibred by surfaces of geometric genus $37$. It gives an answer of a question of Chen and Cui.
Sasaki-Einstein metrics on simply connected rational homology 5-spheres
By developing the method introduced by Kobayashi in 1960’s, Boyer, Galicki and Kollár found many examples of simply connected Sasaki-Einstein 5-manifolds. For such examples they verified existence of orbifold Kaehler-Einstein metrics on various log del Pezzo surfaces via links. By the recent development of method to verify existence of orbifold Kaehler-Einstein metrics, we complete the classification of simply connected rational homology 5-spheres that admits Sasaki-Einstein metrics. This is a joint work with Jihun Park.