Intersecting $ψ$-classes on $M_{0,w}^{\mathrm{trop}}$

Marvin Anas Hahn, Shiyue Li

Published 2021-08-02Version 1

In this paper, we study the intersection products of weighted tropical $\psi$-classes, in arbitrary dimensions, on the moduli space of tropical weighted stable curves. We introduce the tropical Gromov--Witten multiplicity at each vertex of a given tropical curve. This concept enables us to prove that the weight of a maximal cone in an intersection of $\psi$-classes decomposes as the product of tropical Gromov--Witten multiplicities at all vertices of the cone's associated tropical curves. Along the way, we define weighted tropical $\psi$-classes on these moduli spaces, furnish a combinatorial characterisation thereof and realise them as multiples of tropical Weil divisors of a family of rational functions on these moduli spaces. In the special case of top-dimensional intersections, our result confirms the correspondence between the tropical Gromov--Witten invariants and their algebro-geometric counterparts explicitly on these weighted moduli spaces.