Tropicalizations of moduli spaces of curves and covers

在热带几何中，代数变体退化为多面体复合体，称为热带变体。我们把这种退化过程称为热带化。热带几何被特别成功地应用于列举几何中的问题。许多枚举数可以在适当的模空间上用周氏环来表示，以参数化要计数的对象。这在代数几何和热带几何中都成立。一般来说，模空间层次上的联系仍有待理解。在这个提议中，我们计划研究在枚举几何中很重要的几个模空间，即稳定曲线空间的紧化和Hurwitz格式。我们需要研究紧化，以便执行产生枚举数所必需的交点理论。在理想情况下，模空间和它的热带对应物是由热带紧化联系起来的——一个由热带化决定的紧化。在处理高属曲线时，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.