We show that affine cones over smooth cubic surfaces do not admit non-trivial G a -actions.

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-Mustat ¸ȃ and Kaveh-Khovanskii. For this purpose, we define and study special subvarieties, called the Nakayama subvarieties and the positive volume subvarieties, associated to pseudoeffective divisors.

We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an application, we show that they have a Kaehler-Einstein metric if they are general.Comment: 25 page

In a new algebro-geometric way we completely determine whether smooth del Pezzo surfaces are K-(semi)stable or not.

Abstract. For each del Pezzo surface S with du Val singularities, we determine whether it admits a (−KS)-polar cylinder or not. If it allows one, then we present an effective Q-divisor D that is Q-linearly equivalent to −KS and such that the open set S \ Supp(D) is a cylinder. As a corollary, we classify all the del Pezzo surfaces with du Val singularities that admit nontrivial Ga-actions on their affine cones defined by their anticanonical divisors.All considered varieties are assumed to be algebraic and defined over an algebraically closed field of characteristic 0 throughout this article.