On refined count of rational tropical curves

Abstract

We address the problem of existence of refined (i.e., depending on a formal parameter) tropical enumerative invariants, and we present two new examples of a refined count of rational marked tropical curves. One of the new invariants counts plane rational tropical curves with an unmarked vertex of arbitrary valency. It was motivated by the tropical enumeration of plane cuspidal tropical curves given by Y. Ganor and the author, which naturally led to consideration of plane tropical curves with an unmarked fourvalent vertex. Another refined invariant counts rational tropical curves of a given degree in the Euclidean space of arbitrary dimension matching specific constraints, which make the spacial refined invariant similar to known planar invariants.

Tropical curves in abelian surfaces I: enumeration of curves passing through points

“…Such refined invariants have since been extended to many other situations. See for instance [10], [22], [1], [21], [5]. The setting of floor diagrams has also been adapted to enable the computation of some of these refined invariants in [2].…”