Show simple item record

dc.contributor.advisorBaker, Matthew
dc.contributor.authorLuo, Ye
dc.date.accessioned2014-08-27T13:41:44Z
dc.date.available2014-08-27T13:41:44Z
dc.date.created2014-08
dc.date.issued2014-07-02
dc.date.submittedAugust 2014
dc.identifier.urihttp://hdl.handle.net/1853/52323
dc.description.abstractThe divisor theories on finite graphs and metric graphs were introduced systematically as analogues to the divisor theory on algebraic curves, and all these theories are deeply connected to each other via tropical geometry and non-archimedean geometry. In particular, rational functions, divisors and linear systems on algebraic curves can be specialized to those on finite graphs and metric graphs. Important results and interesting problems, including a graph-theoretic Riemann-Roch theorem, tropical proofs of conventional Brill-Noether theorem and Gieseker-Petri theorem, limit linear series on metrized complexes, and relations among moduli spaces of algebraic curves, non-archimedean analytic curves, and metric graphs are discovered or under intense investigations. The content in this thesis is divided into three main subjects, all of which are based on my research and are essentially related to the divisor theory of linear systems on metric graphs and its application to tropical geometry and non-archimedean geometry. Chapter 1 gives an overview of the background and a general introduction of the main results. Chapter 2 is on the theory of rank-determining sets, which are subsets of a metric graph that can be used for the computation of the rank function. A general criterion is provided for rank-determining sets and certain specific examples of finite rank-determining sets are presented. Chapter 3 is on the subject of a tropical convexity theory on linear systems on metric graphs. In particular, the notion of general reduced divisors is introduced as the main tool used to study this tropical convexity theory. Chapter 4 is on the subject of smoothing of limit linear series of rank one on re_ned metrized complexes. A general criterion for smoothable limit linear series of rank 1 is presented and the relations between limit linear series of rank 1 and possible harmonic morphisms to genus 0 metrized complexes are studied.
dc.format.mimetypeapplication/pdf
dc.language.isoen_US
dc.publisherGeorgia Institute of Technology
dc.subjectMetric graph
dc.subjectTropical curves
dc.subjectBerkovich analytification
dc.subjectMetrized complex
dc.subjectLimit linear series
dc.titleLinear systems on metric graphs and some applications to tropical geometry and non-archimedean geometry
dc.typeDissertation
dc.description.degreePh.D.
dc.contributor.departmentMathematics
thesis.degree.levelDoctoral
dc.contributor.committeeMemberYu, Josephine
dc.contributor.committeeMemberRabinoff, Joseph
dc.contributor.committeeMemberLeykin, Anton
dc.contributor.committeeMemberZureick-Brown, David
dc.date.updated2014-08-27T13:41:44Z


Files in this item

Thumbnail

This item appears in the following Collection(s)

Show simple item record