Étale homotopy sections of algebraic varieties

Thesis

We define and study the fundamental pro-finite 2-groupoid of varieties X defined over a field k. This is a higher algebraic invariant of a scheme X, analogous to the higher fundamental path 2-groupoids as defined for topological spaces. This invariant is related to previously defined invariants, for example the absolute Galois group of a field, and Grothendieck’s étale fundamental group.

The special case of Brauer-Severi varieties is considered, in which case a “sections conjecture” type theorem is proved. It is shown that a Brauer-Severi variety X has a rational point if and only if its étale fundamental 2-groupoid has a special sort of section.

Authors: Haydon, J

• Publication date: 2014
• DOI: 10.5287/ora-vy25j8pa4
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: higher-category theory; homotopy theory; arithmetic geometry
• Subjects: Algebraic geometry; Group theory and generalizations (mathematics); Number theory; Algebraic topology