Abstract
We study smooth curves on which the alternating group \(\mathfrak {A}_{6}\) acts faithfully. Let \(\mathcal {V} \subset {\text {PGL}}(3, \mathbb {C})\) be the Valentiner group, which is isomorphic to \(\mathfrak {A}_{6}\). We see that there are integral \(\mathcal {V}\)-invariant curves of degree 12 which have geometric genera 10 and 19. On the other hand, if \(\mathfrak {A}_{6}\) acts faithfully on a curve C of genus 10 or 19, then we give an explicit description of the extension \(k(C / \mathfrak {A}_{5}) / k(C / \mathfrak {A}_{6})\) for any icosahedral subgroup \(\mathfrak {A}_{5}\). Using this, we show the uniqueness of smooth projective curves of genera 10 and 19 whose automorphism groups contain \(\mathfrak {A}_{6}\).
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Acknowledgements
I would like to thank Associate Professor Nobuyoshi Takahashi for detailed advice in this paper.
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Yoshida, Y. \(\mathfrak {A}_{6}\)-invariant curves of genera 10 and 19. Beitr Algebra Geom (2024). https://doi.org/10.1007/s13366-024-00743-0
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DOI: https://doi.org/10.1007/s13366-024-00743-0