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Refined Descendant Invariants of Toric Surfaces

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Abstract

We construct refined tropical enumerative genus zero invariants of toric surfaces that specialize to the tropical descendant genus zero invariants introduced by Markwig and Rau when the quantum parameter tends to 1. In the case of trivalent tropical curves our invariants turn to be the Göttsche–Schroeter refined broccoli invariants. We show that this is the only possible refinement of the Markwig–Rau descendant invariants that generalizes the Göttsche–Schroeter refined broccoli invariants. We discuss also the computational aspect (a lattice path algorithm) and exhibit some examples.

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Notes

  1. Here “rational” means “with rational slopes”.

  2. Here and further on, under the germ we understand a sufficiently small Euclidean neighborhood of the central element.

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Acknowledgements

The authors have been supported by the German–Israeli Foundation Grant No. 1174-197.6/2011 and by the Israel Science Foundation Grants Nos. 176/15 and 501/18, as well as by the Bauer–Neuman Chair in Real and Complex Geometry. This work has been started during the stay of the second author at the Max-Planck Institut für Mathematik, Bonn, in August–September 2015, and then completed during the stay of the second author in the Institute Mittag-Leffler, Stockholm, and École Normale Supérieure, Paris, in 2018. The second author is very grateful to MPIM, IML, and ENS for hospitality and excellent working conditions. We also would like to thank Franziska Schroeter for several important remarks and Travis Mandel for attracting our attention to the work [11]. Special thanks are due to the unknown referee for a careful reading of the paper and making many important critical remarks and suggestions.

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  1. School of Mathematical Sciences, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, 69978, Tel Aviv, Israel

    Lev Blechman & Eugenii Shustin

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  1. Lev Blechman
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  2. Eugenii Shustin
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Correspondence to Eugenii Shustin.

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Blechman, L., Shustin, E. Refined Descendant Invariants of Toric Surfaces. Discrete Comput Geom 62, 180–208 (2019). https://doi.org/10.1007/s00454-019-00093-y

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