Intersections on tropical moduli spaces
Intersections on tropical moduli spaces
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热带模空间上的交点
DOI：
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发表时间：
2008
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通讯作者：
Johannes Rau
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作者：
Johannes Rau
本文探讨了有理后代Gromov-Witten不变量的代数几何理论在多大程度上可以推广到热带世界。尽管我们使用的热带模空间是非紧致的，但答案出人意料地是积极的。我们讨论了弦、因子和伸缩方程，证明了一个描述与“边界”因子相交的分裂引理，并证明了WDVV的一般热带版本。拓扑递归方程(在某些假设下)。作为直接应用，我们证明了环簇$mathbb{P}^1$，$mathbb{P}^2$，$mathbb{P}^1与Mathbb{P}^1$，并且仅结合点条件，热带子孙Gromov-Witten不变量与经典后裔Gromov-Witten不变量重合(推广了Markwig-Rau-2008中$mathbb{P}^2$的结果)。我们的方法使用了热带交集理论，可以统一和简化现有热带计数几何的某些部分(对于有理曲线)。
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 $mathbb{P}^1$, $mathbb{P}^2$, $mathbb{P}^1 imes mathbb{P}^1$ 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).