The major part of this thesis revolves around the real algebraic geometry of curves, especially curves of degree six. We use the topological and rigid isotopy classifications of plane sextics to explore the reality of several features associated to each class, such as the bitangents, inflection points, and tensor eigenvectors. We also study the real tensor rank of plane sextics, the construction of quartic surfaces with prescribed topology, and the avoidance locus, which is the locus of all real lines that do not meet a given plane curve.
In the case of space sextics, a classical construction relates an important family of these genus 4 curves to the del Pezzo surfaces of degree one. We show that this construction simplifies several problems related to space sextics over the field of real numbers. In particular, we find an example of a space sextic with 120 totally real tritangent planes, which answers a historical problem originating from Arnold Emch in 1928.
The last part of this thesis is an algebraic study of a real optimization problem known as Weber problem. We give an explanation and a partial proof for a conjecture on the algebraic degree of the Fermat-Weber point over the field of rational numbers.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:74147 |
| Date | 12 March 2021 |
| Creators | Sayyary Namin, Mahsa |
| Contributors | Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/acceptedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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