Counting Tropical Curves in ℙ1 × ℙ1 : Computation and Polynomiality Properties

Abstract Counts of curves in P^1 P^1 with transverse contact along the zero and infinity sections but fixed contact orders along the zero and infinity fibers have been shown to piecewise polynomial in the entries of the contact order conditions. The result was proved by using tropical methods 36, and in particular the technique of floor diagrams 3. We expand the tropical tools to determine counts of curves in P^1 P^1. We provide a computational tool (building on Polymake 22) that determines such numbers of tropical curves for any genus and any contact orders via a straightforward generalization of Mikhalkin’s lattice path algorithm. The tool can also be used for other toric surfaces. To enable efficient computations by hand, we introduce a new counting tool (for the case of rational curves with transverse contacts with the infinity section) which can be seen as a combination of the floor diagram and the lattice path approach: subfloor diagrams. We use both our computational tool and the subfloor diagrams for experiments revealing structural properties of these counts. We obtain new results on the (piecewise) polynomial structure of counts of rational curves in P^1 P^1 with arbitrary contact orders on the zero and infinity fiber and certain restricted, non-transverse choices for the contact orders on the zero and infinity section.