Explicit desingularisation of Kummer surfaces in characteristic two via specialisation
We study the birational geometry of the Kummer surfaces associated to the Jacobian varieties of genus two curves, with a particular focus on fields of characteristic two. In order to do so, we explicitly compute a projective embedding of the Jacobian of a general genus two curve and, from this, we c...
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Main Author | |
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Format | Journal Article |
Language | English |
Published |
06.09.2024
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Subjects | |
Online Access | Get full text |
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Summary: | We study the birational geometry of the Kummer surfaces associated to the
Jacobian varieties of genus two curves, with a particular focus on fields of
characteristic two. In order to do so, we explicitly compute a projective
embedding of the Jacobian of a general genus two curve and, from this, we
construct its associated Kummer surface. This explicit construction produces a
model for desingularised Kummer surfaces over any field of characteristic not
two, and specialising these equations to characteristic two provides a model of
a partial desingularisation. Adapting the classic description of the Picard
lattice in terms of tropes, we also describe how to explicitly find completely
desingularised models of Kummer surfaces whenever the $p$-rank is not zero. In
the final section of this paper, we compute an example of a Kummer surface with
everywhere good reduction over a quadratic number field, and draw connections
between the models we computed and a criterion that determines when a Kummer
surface has good reduction at two. |
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DOI: | 10.48550/arxiv.2409.04532 |