#### Positivity theory of line bundles and Okounkov bodies = 선속들의 양의 이론과 오쿤코프 체들

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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.
Lee, Yongnamresearcher이용남researcher
Description
한국과학기술원 :수리과학과,
Publisher
한국과학기술원
Issue Date
2019
Identifier
325007
Language
eng
Description

학위논문(박사) - 한국과학기술원 : 수리과학과, 2019.8,[i, 61 p. :]

Keywords

Seshadri constant▼a(extended) okounkov body▼afiltered graded linear series▼ahigher order embeddings; 세샤드리 상수▼a(확장된) 오쿤코프 체▼a필터된 등급 선형체계▼a고차 매입

URI
http://hdl.handle.net/10203/283239