Enumeration of Unicuspidal Curves of Any Degree and Genus on Toric Surfaces

We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative geometry. The same technique applies to enumeration of real plane cuspidal curves: we show that, for any fixed $r\ge 1$ and $d\ge 2r+3$, there exists a generic real $2r$-dimensional linear family of plane curves of degree $d$ in which the number of real $r$-cuspidal curves is asymptotically comparable with the total number of complex $r$-cuspidal curves in the family, as $d\to \infty $.