通过特化显式地对特征为二的Kummer面进行去奇异性处理
Explicit desingularisation of Kummer surfaces in characteristic two via specialisation
AI总结：
本文通过显式构造和特化方法，研究了特征为二的Kummer面的去奇异性处理及良好减少的条件。
摘要：
我们研究了与双曲曲线雅可比簇相关的Kummer面的理化几何，特别关注特征为二的域。为此，我们显式地计算了双曲曲线雅可比簇的射影嵌入，并由此构造其关联的Kummer面。这种显式构造产生了一个在特征不为二的域上去奇异性处理的Kummer面模型，并将这些方程特化到特征为二的域，提供了一个部分去奇异性处理的模型。通过适应经典描述皮卡晶格的方式，我们还描述了如何在p-秩不为零的情况下显式地找到完全去奇异性处理的Kummer面模型。在本文的最后一部分，我们计算了一个在二次数域上处处良好减少的Kummer面，并绘制了我们计算的模型与确定Kummer面在二处良好减少的准则之间的联系。
英文摘要：
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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