Planar tropical cubic curves of any genus, and higher dimensional generalisations

Abstract

A la mémoire de notre ami Jean-Jacques Risler,à qui nous n'avons pas eu le temps de raconter ces incongruités.Abstract. We study the maximal values of Betti numbers of tropical subvarieties of a given dimension and degree in TP n . We provide a lower estimate for the maximal value of the top Betti number, which naturally depends on the dimension and degree, but also on the codimension. In particular, when the codimension is large enough, this lower estimate is larger than the maximal value of the corresponding Hodge number of complex algebraic projective varieties of the given dimension and degree. In the case of surfaces, we extend our study to all tropical homology groups. As a special case, we prove that there exist planar tropical cubic curves of genus g for any non-negative integer g.

Towards tropically counting binodal surfaces

“…When choosing points in Mikhalkin position and a lattice path through the Newton Polytope, we expect to obtain a sliced subdivision corresponding to a floor decomposed surface [BBLdM18]. This decomposition of the surface is used in the description of the surface via tropical floor plans.…”

Harmonic tropical morphisms and approximation

“…converges to the vector (3). Observe that γ j,t has to be oriented coherently with the choice we made to define Res q j .…”

“…. This turns out not to be the case in tropical geometry, as shown in [3]. Consider for instance the tropical curve C ⊂ R 3 of degree 3 and genus 2 given in Fig.…”

Towards Tropically Counting Binodal Surfaces

“…Theorem 5.27 Assuming Conjecture 5.23 is true, it follows that tropical degree d surfaces with a binodal polytope with 6 vertices and width 1 in the dual subdivision Fig. 1 The three families of polytopes with 6 lattice points encoding two nodes at the same vertex for degree d binodal surfaces through points in Mikhalkin position contribute 1 4 d 4 + O(d 3 ) surfaces to the count of binodal degree d surfaces N P 3 2,C (d). So, they contribute to the third-highest term of the polynomial N P 3 2,C (d).…”

“…We show that for small δ, relaxing the notion of a floor plan as defined in [18] to allow nodes in the same or in adjacent floors is not enough to produce the second-order coefficient in Eq. (1). Therefore, surfaces with separated nodes are insufficient to count binodal and trinodal surfaces asymptotically up to two degrees.…”