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We prove that the G\"ottsche-Schroeter and Schroeter-Shustin refined invariants specialize at q=1 to the enumeration of rational, resp. elliptic complex curves on arbitrary toric surfaces matching constraints that consist of points and of points with a contact element. Furthermore, we show that the refined invariant extends to the case of any genus g2 and either one contact constraint or points in Mikhalkin position, and it again specializes to the corresponding characteristic number at q=1. In the appendix we show the limitations of extending this count to a more general setting.
Shustin et al. (Thu,) studied this question.
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