Positivity theory of line bundles and Okounkov bodies

Abstract

This thesis mainly consists of the study of the positivity of line bundles on smooth projective varieties and its connection with convex bodies. In Chapter 2, we study the interaction between the local positivity of line bundles and Okounkov bodies over algebraically closed field of arbitrary characteristic. We also extend the previous theorems on Seshadri constants to graded linear series setting, and introduce the integrated volume function to investigate the relation between Seshadri constants and filtered Okounkov bodies. Moreover, we introduce a convex body of a big divisor, which we call the extended Okounkov body, that is effective in handling the positivity theory associated with multi-point settings. We study their properties, shapes, and describe local positivity data via them. In Chapter 3, we focus on the strong positivity of line bundles. There are several ways to generalize the global generation and very ampleness, which are the classical ways to study the positivity of divisors, and among them, our interests are $k$-very ampleness and higher syzygies on abelian surfaces. First, let $(S,L_{S})$ be a polarized abelian surface, and let $M = c \cdot \pi^*L_S - \alpha \cdot \sum_{i=1}^r E_i$ be a line bundle on ${\rm Bl}_{r}(S)$, where $\pi:{\rm Bl}_{r}(S) \rightarrow S$ is the blow-up of $S$ at $r$ general points with exceptional divisors $E_{1},\dots,E_{r}$. In this setting, we provide a criterion for $k$-very ampleness of $M$, which generalizes the results of Szemberg and Tutaj-Gasi{\'n}ska. Finally, we extend the result of Lazarsfeld-Pareschi-Popa, which connects the Seshadri constants and the higher syzygies of polarized abelian varieties, in the case of abelian surfaces.