Tropical curves with a singularity in a fixed point
Tropical curves with a singularity in a fixed point
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具有固定点奇点的热带曲线
DOI：
10.1007/s00229-011-0471-8
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发表时间：
2009
影响因子：
0.6
通讯作者：
E. Shustin
中科院分区：
文献类型：
--
作者：
H. Markwig;Thomas Markwig;E. Shustin
本文研究了具有不动点奇点的平面曲线族的热带化问题。这种家族的热带化是一种线性热带变种。我们利用线性热带变种的结果刻画了它的最大维锥。我们证明了奇点向高价或更高重数的顶点或向更高权的边的热带化。然后，我们对奇异热带曲线的最大维类型进行了分类。对于这些奇点，奇点要么是两条边的交集，要么是一个重数为3的3叶顶点，或者是权为2的边上的一个点，它到相邻顶点的距离满足一定的度量条件。我们还研究了具有精化热带极限的一般奇异热带曲线，并为它们构造了规范的简单参数化，解释了上述度规条件。
In this paper, we study tropicalisations of families of plane curves with a singularity in a fixed point. The tropicalisation of such a family is a linear tropical variety. We describe its maximal dimensional cones using results about linear tropical varieties. We show that a singularity tropicalises either to a vertex of higher valence or of higher multiplicity, or to an edge of higher weight. We then classify maximal dimensional types of singular tropical curves. For those, the singularity is either a crossing of two edges, or a 3-valent vertex of multiplicity 3, or a point on an edge of weight 2 whose distances to the neighbouring vertices satisfy a certain metric condition. We also study generic singular tropical curves enhanced with refined tropical limits and construct canonical simple parameterisations for them, explaining the above metric condition.