Tropicalizations of moduli spaces of curves and covers

在热带几何中，代数簇退化为多面体复形，称为热带簇。我们把这种退化过程称为热带化。热带几何已特别成功地应用于枚举几何的问题。许多枚举数可以表示在一个合适的模空间参数化的对象计数的周循环。这在代数几何和热带几何中都成立。在模空间水平上的联系仍然有待于一般的理解。在这个建议中，我们计划研究几个模空间，这是重要的枚举几何，即紧致空间的稳定曲线和Hurwitz计划。我们需要研究紧化，以便执行产生枚举数所必需的交集理论。在理想情况下，模空间和它的热带对应物通过热带紧化（tropical compatification）--一种由热带化决定的紧化--联系起来。在处理高亏格曲线时，必须考虑到环面结构和Berkovich分析。

In tropical geometry, algebraic varieties are degenerated to polyhedral complexes called tropical varieties. We refer to this degeneration process as tropicalization. Tropical geometry has been particularly succesfully applied to questions in enumerative geometry. Many enumerative numbers can be expressed in terms of Chow cycles on a suitable moduli space parametrizing the objects to count. This holds true both in algebraic and tropical geometry. A connection at the level of moduli spaces still remains to be understood in general. In this proposal, we plan to study several moduli spaces which are important in enumerative geometry, namely compactifications of spaces of stable curves and Hurwitz schemes. We need to study compactifications in order to perform the intersection theory necessary to produce enumerative numbers. In the ideal case, a moduli space and its tropical counterpart are related by a tropical compactification - a compactification dictated by the tropicalization. When dealing with curves of higher genus, we have to take toroidal structure and Berkovich analytification into account.