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Refined elliptic tropical enumerative invariants

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Abstract

We suggest a new refined (i.e., depending on a parameter) tropical enumerative invariant of toric surfaces. This is the first known enumerative invariant that counts tropical curves of positive genus with marked vertices. Our invariant extends the refined rational broccoli invariant invented by L. Göttsche and the first author, though there is a serious difference between the invariants: our elliptic invariant counts weights assigned partly to individual tropical curves and partly to collections of tropical curves, and our invariant is not always multiplicative over the vertices of the counted tropical curves as was the case for other known tropical enumerative invariants of toric surfaces. As a consequence we define elliptic broccoli curves and elliptic broccoli invariants as well as elliptic tropical descendant invariants for any toric surface.

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Authors and Affiliations

  1. Fachbereich Mathematik (AD), Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Franziska Schroeter

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

    Eugenii Shustin

Authors
  1. Franziska Schroeter
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  2. Eugenii Shustin
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Correspondence to Franziska Schroeter.

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Schroeter, F., Shustin, E. Refined elliptic tropical enumerative invariants. Isr. J. Math. 225, 817–869 (2018). https://doi.org/10.1007/s11856-018-1680-6

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  • DOI: https://doi.org/10.1007/s11856-018-1680-6

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