Tropical pseudostable curves

Abstract

We study the tropical version of the contraction morphism $\mathcal{T}$ between moduli spaces of stable and pseudostable curves. By promoting $\mathcal{T}$ to a logarithmic morphism, we obtain a piecewise linear function between the generalized cone complexes parameterizing tropical stable and pseudostable curves. The ray corresponding to the contracted divisor $\delta_1$ is not contracted to the cone point but mapped onto a ray of $\mathcal{M}_{g,n}^{{\rm trop}, {\rm ps}}$, with a slope reflecting the geometry of the desingularization of a plane cusp. We explore in detail the situation of $g=1$, where the tautological geometry of both spaces is fully described by piecewise polynomial functions on the tropical moduli spaces.