원준영 교수

We show that affine cones over smooth cubic surfaces do not admit non-trivial \mathbb{G}_{a} -actions.

For each del Pezzo surface $S$ with du Val singularities, we determine whether it admits a $(-K_S)$-polar cylinder or not. If it allows one, then we present an effective $\mathbb{Q}$-divisor $D$ that is $\mathbb{Q}$-linearly equivalent to $-K_S$ and such that the open set $S\setminus\mathrm{Supp}(D)$ is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit nontrivial $\mathbb{G}_a$-actions on their affine cones defined by their anticanonical divi

On del Pezzo surfaces, we study effective ample |$\mathbb{R}$|-divisors such that the complements of their supports are isomorphic to |$\mathbb{A}^1$|-bundles over smooth affine curves. All considered varieties are assumed to be algebraic and defined over an algebraically closed field of characteristic |$0$| throughout this article.

An Okounkov body is a convex subset in Euclidean space associated to a big divisor on a smooth projective variety with respect to an admissible flag. In this paper, we introduce two convex bodies associated to pseudoeffective divisors, called the valuative Okounkov bodies and the limiting Okounkov bodies, and show that these convex bodies reflect the asymptotic properties of pseudoeffective divisors as in the case with big divisors. Our results extend the works of Lazarsfeld–Mustaţă and Kaveh–Kh

We compute global log canonical thresholds, or equivalently alpha invariants, of birationally rigid Fano three-folds embedded in weighted projective spaces as codimension two or three. As an important application, we prove that most of them are weakly exceptional, |$K$|-stable and admit Kähler–Einstien metric.

Abstract We compute the global log-canonical thresholds (lct) of del Pezzo surfaces of degrees ≥ 2 with du Val singularities.

Abstract We classify all the effective anticanonical divisors on weak del Pezzo surfaces. Through this classification we obtain the smallest number among the log canonical thresholds of effective anticanonical divisors on a given Gorenstein canonical del Pezzo surface.

We completely determine which simply connected rational homology 5-spheres admit Sasaki-Einstein metrics.

We first prove some basic properties of Okounkov bodies, and give a characterization of Nakayama and positive volume subvarieties of a pseudoeffective divisor in terms of Okounkov bodies. Next, we show that each valuative and limiting Okounkov bodies of a pseudoeffective divisor which admits the birational good Zariski decomposition is a rational polytope with respect to some admissible flag. This is an extension of the result of Anderson-Küronya-Lozovanu about the rational polyhedrality of Okou

We give examples of K-unstable singular del Pezzo surfaces which are weighted hypersurfaces with index 2.