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公式化，但仍然很难回答，特别是当工作在不同于复数的领域时。这种内在的复杂性促成了枚举几何在过去几十年的蓬勃发展，在数学的各个领域，如代数几何，算术几何，表示论，数学物理，随机矩阵理论，甚至理论物理和弦理论之间提供了富有成效的联系。近年来，所谓的精化枚举不变量作为算术几何中的一种计数方法，为几何对象的计数提供了一种普遍的理论。有了这个建议，我建议，以丰富理论的算术计数引入一个新的工具，这一理论：退化和tropicalizations.Tropical几何工具是一个现代领域，它允许代数几何和组合学之间的方法交换。通过一个退化过程称为热带化，一个代数簇变成一个热带簇。后者是满足一定条件的多面体复形。热带几何提供了许多其他领域的数学连接，如辛几何，算术几何，数学物理和优化，但也领域的应用数学，如经济，机器学习和计算生物学。在2002年，以下建议的孔采维奇，米哈尔金开始使用热带方法在枚举几何证明著名的米哈尔金对应定理。该项目将集中在算术计数的双切线光滑平面四次曲线。这是一个自然的出发点，作为理论的热带双切线的热带四次是很好地理解，不仅如果我们tropicalize超过复数，但也如果我们tropicalize超过真实的数字。已经Plücker知道，一个平面四次曲线有28双切线线的复数。然而，在实数上，四次曲线可以有4、8、16或28个双切线，这取决于真实的曲线的拓扑类型。最近，Larson和沃格特提出了一种双切线的算术计数方法，当该方法专门用于真实的数时，可以得到一个在某些情况下不变的符号计数，类似于著名的满足点条件的d次有理平面曲线的Welschinger符号计数。我们还将这个回归线数与立方曲面中的回归线联系起来。作为一个长期的观点，我们研究算术计数的平面曲线满足点条件。

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.