We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is (Tyomkin in Adv Math 305:1356–1383, 2017), where a tropical-algebraic correspondence theorem was proved that relates counts of rational curves in toric varieties that satisfy point conditions and cross-ratio constraints to the analogous tropical counts. We proceed in two steps: based on tropical intersection theory we first study tropical cross-ratios and introduce degenerated cross-ratios. Second we provide a lattice path algorithm that produces all rational tropical curves satisfying such degenerated conditions explicitly. In a special case simpler combinatorial objects, so-called cross-ratio floor diagrams, are introduced which can be used to determine these enumerative numbers as well.

中文翻译：

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计算具有交叉比约束的热带有理曲线

我们使用热带和组合方法枚举通过点并满足交叉比约束的复曲面中的有理曲线。我们的起点是（Tyomkin in Adv Math 305:1356–1383, 2017），其中证明了热带代数对应定理，该定理将满足点条件和交叉比约束的复曲面变体中的有理曲线计数与类似的热带计数联系起来. 我们分两步进行：基于热带交叉理论，我们首先研究热带交叉比并引入退化交叉比。其次，我们提供了一种格子路径算法，该算法可生成明确满足此类退化条件的所有有理热带曲线。在特殊情况下，更简单的组合对象，即所谓的交叉比楼层图，