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

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 $g$.