Curves of genus 2 with group of automorphisms isomorphic to or |
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Authors: | Gabriel Cardona Jordi Quer |
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Affiliation: | Departament Ciències Matemàtiques i Inf., Universitat de les Illes Balears, Ed. Anselm Turmeda, Campus UIB, Carretera Valldemossa, km. 7.5, E-07122 -- Palma de Mallorca, Spain ; Departament Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Ed. Omega, Campus Nord, Jordi Girona, 1-3, E-08034 -- Barcelona, Spain |
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Abstract: | The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety. In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set . The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class. |
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Keywords: | Curves of genus $2$ twists of curves |
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