arXiv 0812.36782008-12math.AG

热带模空间上的交

Intersections on tropical moduli spaces

Johannes Rau

备注：: 40 pages, 17 Postscript figures; updated to fit the published version
发表信息：: Rocky Mt. J. Math. 46.2 (2016), pp. 581-662
DOI：: 10.1216/RMJ-2016-46-2-581

摘要：

本文探讨在何种程度上代数几何理论的合理后裔Gromov-Witten不变量可以结转到热带世界。尽管我们研究的热带模空间是非紧的，但答案是令人惊讶的肯定的。我们讨论了字符串，除数和Eklion方程，我们证明了分裂引理描述的交集与“边界”除数，我们证明了一般热带版本的WDVV分别。拓扑递归方程（在某些假设下）。作为一个直接的应用，我们证明了环面品种$\mathbb{P}^1$，$\mathbb{P}^2$，$\mathbb{P}^1 \times \mathbb{P}^1$和Psi条件仅与点条件相结合，热带和经典的后裔Gromov-Witten不变量一致（这扩展了Markwig-Rau-2008中的$\mathbb{P}^2$的结果）。我们的方法使用热带相交理论，可以统一和简化现有的热带枚举几何的某些部分（有理曲线）。

英文摘要：

This article explores to which extent the algebro-geometric theory of rational descendant Gromov-Witten invariants can be carried over to the tropical world. Despite the fact that the tropical moduli-spaces we work with are non-compact, the answer is surprisingly positive. We discuss the string, divisor and dilaton equations, we prove a splitting lemma describing the intersection with a "boundary" divisor and we prove general tropical versions of the WDVV resp. topological recursion equations (under some assumptions). As a direct application, we prove that the toric varieties $\mathbb{P}^1$, $\mathbb{P}^2$, $\mathbb{P}^1 \times \mathbb{P}^1$ and with Psi-conditions only in combination with point conditions, the tropical and classical descendant Gromov-Witten invariants coincide (which extends the result for $\mathbb{P}^2$ in Markwig-Rau-2008). Our approach uses tropical intersection theory and can unify and simplify some parts of the existing tropical enumerative geometry (for rational curves).

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