Sung Rak Choi

Geography of log models: theory and applications

This is an introduction to geography of log models with applications to positive cones of Fano type (FT) varieties and to geometry of minimal models and Mori fibrations.

Potentially non-klt locus and its applications

We introduce the notion of potentially klt pairs for normal projective varieties with pseudoeffective anticanonical divisor. The potentially non-klt locus is a subset of X which is birationally transformed precisely into the non-klt locus on a

Geography of log models via asymptotic base loci

-K_X

Duality of the cones of divisors and curves

-KX-minimal model of X. We prove basic properties of potentially non-klt locus in comparison with those of classical non-klt locus. As applications, we give a new characterization of varieties of Fano type, and we also improve results on the rational connectedness of uniruled varieties with pseudoeffective anticanonical divisor.

Asymptotic base loci via Okounkov bodies

An Okounkov body is a convex body in Euclidean space associated to a big divisor on a smooth projective variety with respect to an admissible flag. We introduce two different convex bodies associated to pseudoeffective divisors, called the valuative Okounkov bodies and limiting Okounkov bodies. As in the case with big divisors, these convex bodies reflect asymptotic properties of pseudoeffective divisors.

The geography of log models and its applications

The geography of log models refers to the decomposition of the set of effective adjoint divisors into the polytopes defined by the resulting models obtained by the log minimal model program. We will describe the geography of log models in terms of the asymptotic base loci and Zariski decompositions of adjoint divisors. As an application, we prove some structure theorems on partially ample cones, thereby giving a partial answer to a question of B. Totaro.