Tropical Hurwitz loci

枚举几何是代数几何的一部分，其中计算满足某些条件的几何对象，例如通过指定重数的给定点的平面曲线。这些问题通常很容易公式化，但很难用基本技术解决。一个经常成功的方法是将一个枚举几何问题转化为一个适当的模空间上的交集问题。这就是为什么枚举几何在推动代数几何的新发展和与其他领域如辛几何或弦理论的联系方面非常富有成效的原因。热带几何是一个最近发展迅速的领域，其中代数几何对象退化为某些称为热带簇的分段线性组合对象。尽管退化，许多代数几何不变量结转到热带世界。在这些情况下，热带几何是研究代数几何问题的有效计算工具。它也与其他领域如最优化或生物数学有联系。热带几何在计数问题的研究中取得了特别的成功。热带枚举几何领域是由Mikhalkin开创的，他的著名对应定理将经典的平面曲线数与热带对应数联系起来。这一建议研究枚举几何和热带几何相互作用的问题，旨在获得两个领域的结果，并更深入地了解它们之间的联系。感兴趣的主要对象是被称为（双）Hurwitz轨迹的射影线的分支覆盖的轨迹。一个0维Hurwitz轨迹经典地称为Hurwitz数，Cavalieri，约翰逊和我已经引入了双Hurwitz数的热带类似物，双Hurwitz数在固定分歧数据的项中具有有趣的分段多项式结构.热带的方法是有帮助的，发现新的功能，这种分段多项式structure.In这个建议，我建议延长图片到高维循环的模空间中的曲线参数化某些分歧覆盖。我打算研究-古典和热带-赫维茨轨迹，他们的分段多项式结构和跨壁公式。双轨的方法-古典和热带-服务于两个目的：第一，我们可以使用这两种方法的整个机器，以获得最大的结果。其次，通过对赫尔维茨轨迹理论的两个方面的深入研究，希望能进一步了解古典几何与热带几何之间的联系。此外，赫尔维茨轨迹的研究及其与热带对应物的联系将推进热带模空间及其相交理论的研究，并可能有助于对双赫尔维茨轨迹的跨壁公式有更多的几何理解。

Enumerative geometry is a part of algebraic geometry in which one counts geometric objects satisfying certain conditions, e.g. plane curves passing through given points with assigned multiplicities. These questions are typically easy to formulate, but hard to solve with elementary techniques. An often succesful approach consists in translating an enumerative geometric question into an intersection problem on some appropriate moduli space. This is a reason why enumerative geometry has been very fruitful in pushing new developments in algebraic geometry and in making connections to other fields such as symplectic geometry or string theory.Tropical geometry is a recent and quickly growing field in which algebro-geometric objects are degenerated to certain piecewise linear combinatorial objects called tropical varieties. In spite of the degeneration, many algebro-geometric invariants carry over to the tropical world. In these cases tropical geometry is an effective computational tool to study problems in algebraic geometry. It also has connections to other fields such as optimization or biomathematics.Tropical geometry had particular success in the study of enumerative questions. The field of tropical enumerative geometry was pioneered by Mikhalkin with his celebrated Correspondence Theorem relating classical numbers of plane curves to their tropical counterparts.This proposal suggests to study questions at the interplay of enumerative geometry and tropical geometry, aiming at results in both fields as well as at a deeper understanding of their connections. The main object of interest are loci of ramified covers of the projective line called (double) Hurwitz loci. A 0-dimensional Hurwitz locus is classically known as Hurwitz number.Tropical analogues of double Hurwitz numbers have been introduced by Cavalieri, Johnson and myself.Double Hurwitz numbers have an interesting piecewise polynomial structure in the entries of the fixed ramification data. The tropical approach was helpful to discover new features of this piecewise polynomial structure.In this proposal, I suggest to extent the picture to higher-dimensional cycles in the moduli space of curves parametrizing certain ramified covers. I intend to study - classical and tropical - Hurwitz loci, their piecewise polynomial structure and wall-crossing formulas.The double-tracked approach - classical and tropical - serves two purposes: first, we can use the whole machinery of both methods to gain maximal outcome. Second, I hope to learn more about the connection between classical and tropical geometry by thouroughly investigating the two sides of the theory of Hurwitz loci. In addition, the study of Hurwitz loci and their connections to their tropical counterparts will advance the study of tropical moduli spaces and their intersection theory in general and might contribute to a more geometric understanding of the wall-crossing formulas for double Hurwitz loci.