Planar Tropical Cubic Curves of Any Genus, and Higher Dimensional Generalisations

TL;DR

This paper investigates the Betti numbers of tropical subvarieties, providing lower bounds that surpass complex algebraic counterparts in certain cases, and constructs tropical cubic curves of any genus.

Contribution

It offers new lower estimates for Betti numbers of tropical varieties and demonstrates the existence of tropical cubic curves with arbitrary genus.

Findings

Lower bounds for top Betti numbers depend on dimension, degree, and codimension.

In certain cases, these bounds exceed complex algebraic Hodge numbers.

Existence of planar tropical cubic curves of any genus g.

Abstract

We study the maximal values of Betti numbers of tropical subvarieties of a given dimension and degree in $\mathbb{TP}^n$. 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 $g$.