Irreducible cycles and points in special position in moduli spaces for tropical curves

TL;DR

This paper investigates the irreducibility of cycles in moduli spaces of tropical curves and characterizes special position loci of point configurations as tropical cycles with explicit weights.

Contribution

It establishes irreducibility results for Psi-classes and vital divisors, and constructs a tropical cycle structure for special position loci in point configurations.

Findings

Psi-classes and vital divisors are irreducible

Locally irreducible divisors are globally irreducible for n ≤ 6

Explicit weights of the tropical cycle for special position loci are computed

Abstract

In the first part of this paper, we discuss the notion of irreducibility of cycles in the moduli spaces of n-marked rational tropical curves. We prove that Psi-classes and vital divisors are irreducible, and that locally irreducible divisors are also globally irreducible for n \leq 6. In the second part of the paper, we show that the locus of point configurations in (\R^2)^n in special position for counting rational plane curves (defined in two different ways) can be given the structure a tropical cycle of codimension 1. In addition, we compute explicitly the weights of this cycle.