Lifting harmonic morphisms II: tropical curves and metrized complexes
Lifting harmonic morphisms II: tropical curves and metrized complexes
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提升调和态射 II：热带曲线和度量复形
DOI：
10.2140/ant.2015.9.267
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Joseph Rabinoff
中科院分区：
文献类型：
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作者：
O. Amini;M. Baker;Erwan Brugall'e;Joseph Rabinoff
在本文中，我们证明了热带曲线态射的几个提升定理。我们将增广度量图的有限调和态射提升为代数曲线态射的障碍解释为某些赫尔维茨数的不消失，并且我们给出了这种障碍消失的各种条件。特别是，我们证明（非增广）度量图的任何有限调和态射都会提升。我们还给出了这些结果的各种应用。例如，我们证明热带曲线 C 上除数的线性等价性与通过声明从 C 到热带射影线的每个有限调和态射的纤维是等价的而生成的等价关系一致。我们研究配备有限群作用的度量复合体的可提升性，并用它来对作为超椭圆曲线的热带化而出现的所有增强度量图进行分类。我们证明存在一条 d 角热带曲线，它不会提升为 d 角代数曲线。
本文是系列文章中的第二篇。
In this paper we prove several lifting theorems for morphisms of tropical curves. We interpret the obstruction to lifting a finite harmonic morphism of augmented metric graphs to a morphism of algebraic curves as the non-vanishing of certain Hurwitz numbers, and we give various conditions under which this obstruction does vanish. In particular we show that any finite harmonic morphism of (non-augmented) metric graphs lifts. We also give various applications of these results. For example, we show that linear equivalence of divisors on a tropical curve C coincides with the equivalence relation generated by declaring that the fibers of every finite harmonic morphism from C to the tropical projective line are equivalent. We study liftability of metrized complexes equipped with a finite group action, and use this to classify all augmented metric graphs arising as the tropicalization of a hyperelliptic curve. We prove that there exists a d-gonal tropical curve that does not lift to a d-gonal algebraic curve.
This article is the second in a series of two.