Topology of tropical moduli spaces of weighted stable curves in higher genus

Abstract

Given integers g ≥ 0, n ≥ 1, and a vector w ∈ (Q ∩ (0, 1]) n such that 2g − 2 + w i > 0, we study the topology of the moduli space ∆ g,w of w-stable tropical curves of genus g with volume 1. The space ∆ g,w is the dual complex of the divisor of singular curves in Hassett's moduli space of w-stable genus g curves M g,w . When g ≥ 1, we show that ∆ g,w is simply connected for all values of w. We also give a formula for the Euler characteristic of ∆ g,w in terms of the combinatorics of w.

Wonderful compactifications and rational curves with cyclic action

“…to study intersection numbers on L 𝑟 𝑛 (along the lines of [Kat12, KM09,HL22]). We may also study the reduced rational cohomology of the locus of tropical curves with total edge length 1 in 𝐿 𝑟 ,trop 𝑛 to understand the mixed Hodge structure of L 𝑟 𝑛 , in the sense of [Del71,Del74] and along the lines of [CGP21,KLSY20]. This is made possible by the observation that the boundary L…”

Automorphisms of tropical Hassett spaces

“…Closed formulas for the number of spheres are known when w is heavy/light. In higher genus, the topology of g;w has been explored by Li, Serpente, Yun, and the third author in [19], and by Serpente in [27]. When g 1, and for any value of w, the space g;w is shown to be simply-connected.…”

“…There is a geometric realization functor given by X 7 ! jX j; see [9,17,19] for a description of this functor.…”