Arithmetic counts of bitangents to plane quartics by means of tropical geometry

计数几何是几何计数问题的研究领域。许多枚举问题可以很容易地表述出来，但仍然很难回答，特别是在处理与复数不同的字段时。在过去的几十年里，这种内在的复杂性促成了枚举几何的蓬勃发展，为代数几何、算术几何、表示理论、数学物理、随机矩阵理论甚至理论物理和弦理论等各个数学领域提供了富有成效的联系。近年来，作为算术几何中的一种计数方法，所谓的精炼计数不变量提供了一种对几何对象进行计数的通用理论。有了这个建议，我建议通过引入一个新的工具来丰富算术计数理论：退化和热带化工具。热带几何是一个现代领域，它允许在代数几何和组合几何之间交换方法。通过称为热带化的退化过程，代数品种变成热带品种。后者是满足一定条件的多面体配合物。热带几何为数学中的许多其他领域提供了联系，如辛几何、算术几何、数学物理和优化，但也为数学应用领域的领域，如经济学、机器学习和计算生物学提供了联系。2002年，根据Kontsevich的建议，米哈尔金通过证明著名的米哈尔金对应定理，开始在枚举几何中使用热带方法。该项目将侧重于算法计数的bitangents平滑平面四次曲线。这是一个很自然的起点，因为热带四次元的热带双线理论已经很好地理解了，不仅在复数上进行热带化，而且在实数上也进行热带化。plicker已经知道平面四次曲线在复数上有28条切线。然而，在实数曲线上，根据实数曲线的拓扑类型，一个四次曲线可以有4、8、16或28个点。最近，Larson和Vogt提出了一种bitangents的算术计数方法，当将其专门用于实数时，会得到一个在某些情况下不变的符号计数，类似于著名的Welschinger d次满足点条件的有理平面曲线的符号计数。在这个项目中，我们计划提供工具，以热带几何方法计算bitangents的算术多重性。我们还把这个热带数与一个立方曲面上的热带线联系起来。从长远的角度来看，我们研究了满足点条件的平面曲线的算术计数。

Enumerative geometry is the research area of geometric counting problems. Many enumerative questions can be formulated easily, but remain notoriously hard to answer, in particular when working over a field different from the complex numbers. This intrinsic complexity has contributed to the flourishing of enumerative geometry over the last decades, providing fruitful connections among various areas of mathematics such as algebraic geometry, arithmetic geometry, representation theory, mathematical physics, the theory of random matrices and even to theoretical physics and string theory. Recently, so-called refined enumerative invariants which can be viewed as a counting method in arithmetic geometry offer a universal theory of counting geometric objects. With this proposal, I suggest to enrich the theory of arithmetic counting by introducing a new tool to this theory: the tool of degenerations and tropicalizations.Tropical geometry is a modern area which allows an exchange of methods between algebraic geometry and combinatorics. Through a degeneration process called tropicalization, an algebraic variety is turned into a tropical variety. The latter is a polyhedral complex satisfying certain conditions. Tropical geometry provides connections to many other areas within mathematics, such as symplectic geometry, arithmetic geometry, mathematical physics and optimization, but also to areas in fields of application of mathematics such as economy, machine learning and computational biology.In 2002, following suggestions of Kontsevich, Mikhalkin initiated the use of tropical methods in enumerative geometry by proving the celebrated Mikhalkin correspondence theorem. The project will focus on arithmetic counts of bitangents to smooth plane quartic curves. This is a natural starting point, as the theory of tropical bitangents to a tropical quartic is well understood, not only if we tropicalize over the complex numbers but also if we tropicalize over the real numbers.Already Plücker knew that a plane quartic curve has precisely 28 bitangent lines over the complex numbers. Over the reals however, a quartic can have 4, 8, 16 or 28 bitangents, depending on the topological type of the real curve. Quite recently, Larson and Vogt initiated an arithmetic count of bitangents, which, when specialized to the real numbers, yields a signed count which is invariant under some circumstances, similar to the famous Welschinger signed count of rational plane curves of degree d satisfying point conditions.In this project, we plan to provide tools to compute arithmetic multiplicities of bitangents by means of tropical geometry. We also relate this tropical count to the tropical lines in a cubic surface. As a long-term perspective, we investigate arithmetic counts of plane curves satisfying point conditions.