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Group actions on curves over arbitrary fields Garcia Armas, Mario


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 G-action 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 Gʹ-action that lifts the prescribed G-action on the projective line. Alternatively, this problem may be regarded as the Galois embedding problem given by the surjection Gʹ ↠ G and the G-Galois 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 G-curve defined over k. Recall that a G-curve is an algebraic curve endowed with the action of a finite group G. A faithful G-curve C is called strongly incompressible if every dominant G-equivariant rational map of C onto a faithful G-variety is birational. We prove that strongly incompressible G-curves 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 G-curves depends on finer arithmetic properties of k.

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