Tropical Singularities

在热带几何中，代数簇退化为多面体复形，称为热带簇。尽管退化，代数簇的许多性质可以从它的热带对应物中读出，并且许多定理在热带方面继续成立。因此，热带几何提供了一种方法来研究问题的代数几何的手段，离散数学和组合学。热带几何已成功地应用于计算问题和枚举几何问题，特别是真实的枚举问题。要真正地利用好这一新工具，就必须回答许多翻译本质上的问题。这个项目的中心问题是：什么是奇点的概念的正确翻译到热带世界？这个问题是很自然的问，因此最近有几个作者感兴趣（[3]，[4]，[13]）。一般来说，这个问题很难回答，这一事实使它更加耐人寻味。该项目的目的是大大提高热带奇点的理解，将结果应用于枚举几何的问题，并调查这一主题与组合学问题的联系。

In tropical geometry, algebraic varieties are degenerated to polyhedral complexes called tropical varieties. In spite of the degeneration, many properties of an algebraic variety can be read off its tropical counterpart, and many theorems continue to hold on the tropical side. Tropical geometry thus provides an approach to study questions in algebraic geometry by means of discrete mathematics and combinatorics. Tropical geometry has been successfully applied to computational problems and problems in enumerative geometry, in particular real enumerative questions. To really make good use of the new tool, many questions of translating nature have to be answered. The central question for this project is: what is the correct translation of the concept of a singularity to the tropical world? This question is quite natural to ask and has consequently interested several authors recently ([3], [4], [13]). The fact that it is hard to answer in general makes it even more intriguing. The aim of this project is to considerably improve the understanding of tropical singularities, to apply the results to questions in enumerative geometry, and to investigate the connections of this subject to questions in combinatorics.