The Realizability of Curves in a Tropical Plane

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• 作者：Anna Lena Birkmeyer ; Andreas Gathmann ; Kirsten Schmitz
• 关键词：Tropical geometry ; Tropicalization ; Tropical realizability
• 刊名：Discrete & Computational Geometry
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：57
• 期：1
• 页码：12-55
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Combinatorics; Computational Mathematics and Numerical Analysis;
• 出版者：Springer US
• ISSN：1432-0444
• 卷排序：57

文摘

Let E be a plane in an algebraic torus \( (K^*)^n \) over an algebraically closed field K. Given a balanced 1-dimensional fan C in the tropicalization of E, i. e. in the Bergman fan of the corresponding matroid, we give a complete algorithmic answer to the question whether or not C can be realized as the tropicalization of an algebraic curve contained in E. Moreover, in the case of realizability the algorithm also determines the dimension of the moduli space of all algebraic curves in E tropicalizing to C, a concrete simple example of such a curve, and whether C can also be realized by an irreducible algebraic curve in E. In the first important case when E is a general plane in a 3-dimensional torus we also use our algorithm to prove some general criteria for C that imply its realizability resp. non-realizability. They include and generalize the main known obstructions by Brugallé-Shaw and Bogart-Katz coming from tropical intersection theory.