Pseudotropical curves

Sergei Lanzat; Michael Polyak

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arXiv · 1812.00248

Pseudotropical curves

Sergei Lanzat·Michael Polyak

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Abstract

We propose a generalization of tropical curves by dropping the rationality and integrality requirements while preserving the balancing condition. An interpretation of such curves as critical points of a certain quadratic functional allows us to settle the existence and uniqueness problem. The machinery of dual polygons and the intersection theory also generalize as expected. We study the homology of a compactified moduli space of rigid oriented marked curves. A weighted count of rational pseudotropical curves passing through a generic collection of points is interpreted via top-degree cycles on the moduli. We construct a family of such cycles using quantum tori Lie algebras and show that in the usual tropical case this gives the refined curve count of Block and G\"ottsche. Finally, we derive a recursive formula for this Lie-weighted count of rational pseudotropical curves.

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BibTeX RIS

Sergei Lanzat, Michael Polyak. 2018-12-01. Pseudotropical curves. https://arxiv.org/abs/1812.00248

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