论文标题
Bitangents通过热带几何形状到平面四分之一
Bitangents to plane quartics via tropical geometry: rationality, $\mathbb{A}^1$-enumeration, and real signed count
论文作者
论文摘要
我们探索了热带方法的扩展，以列举枚举问题，例如$ \ mathbb {a}^1 $ - 增生，并具有Grothendieck-Witt环中的值，以及使用Bitangents将Bitangents用于平面Quartsics作为验证案例的Henselian Reality Fields的合理性。我们考虑在热带化的光滑且满足轻度通用条件的有价值的田地上的四分之一曲线。然后，我们在热带化边缘扭曲边缘方面表达了比特安格斯的理性及其相切点的障碍。后者仅取决于定义方程模量正方形的热带化和初始系数。我们还表明，热带bitangent的GW-Multiplicition，即，其升力的多重性有助于$ \ m athbb {a}^1 $ - Larson和Vogt定义的Bitangents的数量，可以从四重奏的热带化以及最初的系数以及定义等式的最初系数中计算出来。作为一个应用程序，我们表明，大多数热带bitangent类的四个升降机贡献了$ 2 \ mathbb {h} $，两倍是双曲机平面的两倍，到$ \ mathbb {a}^1 $ enumeration。这些结果依赖于将Henselian估算场的Grothendieck-Witt环与其残留场的Grothendieck-Witt环相关的变性定理，其残基特性不等于两个。
We explore extensions of tropical methods to arithmetic enumerative problems such as $\mathbb{A}^1$-enumeration with values in the Grothendieck-Witt ring, and rationality over Henselian valued fields, using bitangents to plane quartics as a test case. We consider quartic curves over valued fields whose tropicalizations are smooth and satisfy a mild genericity condition. We then express obstructions to rationality of bitangents and their points of tangency in terms of twisting of edges of the tropicalization; the latter depends only on the tropicalization and the initial coefficients of the defining equation modulo squares. We also show that the GW-multiplicity of a tropical bitangent, i.e., the multiplicity with which its lifts contribute to the $\mathbb{A}^1$-enumeration of bitangents as defined by Larson and Vogt, can be computed from the tropicalization of the quartic together with the initial coefficients of the defining equation. As an application, we show that the four lifts of most tropical bitangent classes contribute $2\mathbb{H}$, twice the class of the hyperbolic plane, to the $\mathbb{A}^1$-enumeration. These results rely on a degeneration theorem relating the Grothendieck-Witt ring of a Henselian valued field to the Grothendieck-Witt ring of its residue field, in residue characteristic not equal to two.
