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 Group actions on curves over arbitrary fields
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Group actions on curves over arbitrary fields Garcia Armas, Mario
Abstract
This thesis consists of three parts. The common theme is finite group actions on algebraic curves defined over an arbitrary field k. In Part I we classify finite group actions on irreducible conic curves defined over k. Equivalently, we classify finite (constant) subgroups of SO(q) up to conjugacy, where q is a nondegenerate quadratic form of rank 3 defined over k. In the case where k is the field of complex numbers, these groups were classified by F. Klein at the end of the 19th century. In recent papers of A. Beauville and X. Faber, this classification is extended to the case where k is arbitrary, but q is split. We further extend their results by classifying finite subgroups of SO(q) for any base field k of characteristic ≠ 2 and any nondegenerate ternary quadratic form q. In Part II we address the Hyperelliptic Lifting Problem (or HLP): Given a faithful Gaction on ℙ¹ defined over k and an exact sequence 1 → μ₂ → Gʹ→ G → 1, determine the conditions for the existence of a hyperelliptic curve C/k endowed with a faithful Gʹaction that lifts the prescribed Gaction on the projective line. Alternatively, this problem may be regarded as the Galois embedding problem given by the surjection Gʹ ↠ G and the GGalois extension k(ℙ¹)/k(ℙ¹)G. In this thesis, we find a complete solution to the HLP in characteristic 0 for every faithful group action on ℙ¹ and every exact sequence as above. In Part III we determine whether, given a finite group G and a base field k of characteristic 0, there exists a strongly incompressible Gcurve defined over k. Recall that a Gcurve is an algebraic curve endowed with the action of a finite group G. A faithful Gcurve C is called strongly incompressible if every dominant Gequivariant rational map of C onto a faithful Gvariety is birational. We prove that strongly incompressible Gcurves exist if G cannot act faithfully on the projective line over k. On the other hand, if G does embed into PGL₂ over k, we show that the existence of strongly incompressible Gcurves depends on finer arithmetic properties of k.
Item Metadata
Title 
Group actions on curves over arbitrary fields

Creator  
Publisher 
University of British Columbia

Date Issued 
2015

Description 
This thesis consists of three parts. The common theme is finite group actions on algebraic curves defined over an arbitrary field k.
In Part I we classify finite group actions on irreducible conic curves defined over k. Equivalently, we classify finite (constant) subgroups of SO(q) up to conjugacy, where q is a nondegenerate quadratic form of rank 3 defined over k. In the case where k is the field of complex numbers, these groups were classified by F. Klein at the end of the 19th century. In recent papers of A. Beauville and X. Faber, this classification is extended to the case where k is arbitrary, but q is split. We further extend their results by classifying finite subgroups of SO(q) for any base field k of characteristic ≠ 2 and any nondegenerate ternary quadratic form q.
In Part II we address the Hyperelliptic Lifting Problem (or HLP): Given a faithful Gaction on ℙ¹ defined over k and an exact sequence 1 → μ₂ → Gʹ→ G → 1, determine the conditions for the existence of a hyperelliptic curve C/k endowed with a faithful Gʹaction that lifts the prescribed Gaction on the projective line. Alternatively, this problem may be regarded as the Galois embedding problem given by the surjection Gʹ ↠ G and the GGalois extension k(ℙ¹)/k(ℙ¹)G. In this thesis, we find a complete solution to the HLP in characteristic 0 for every faithful group action on ℙ¹ and every exact sequence as above.
In Part III we determine whether, given a finite group G and a base field k of characteristic 0, there exists a strongly incompressible Gcurve defined over k. Recall that a Gcurve is an algebraic curve endowed with the action of a finite group G. A faithful Gcurve C is called strongly incompressible if every dominant Gequivariant rational map of C onto a faithful Gvariety is birational. We prove that strongly incompressible Gcurves exist if G cannot act faithfully on the projective line over k. On the other hand, if G does embed into PGL₂ over k, we show that the existence of strongly incompressible Gcurves depends on finer arithmetic properties of k.

Genre  
Type  
Language 
eng

Date Available 
20150323

Provider 
Vancouver : University of British Columbia Library

Rights 
AttributionNonCommercialNoDerivs 2.5 Canada

DOI 
10.14288/1.0135694

URI  
Degree  
Program  
Affiliation  
Degree Grantor 
University of British Columbia

Graduation Date 
201505

Campus  
Scholarly Level 
Graduate

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Aggregated Source Repository 
DSpace

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Rights
AttributionNonCommercialNoDerivs 2.5 Canada