The Moduli Space of Harnack Curves in Toric Surfaces

Abstract

In 2006, Kenyon and Okounkov computed the moduli space of Harnack curves of degree $d$ in $\mathbb{C}{\mathbb{P}}^2$. We generalize to any projective toric surface some of the techniques used there. More precisely, we show that the moduli space ${\mathcal{H}}_{\Delta }$ of Harnack curves with Newton polygon $\Delta$ is diffeomorphic to ${\mathbb{R}}^{m-3}\times {\mathbb{R}}_{\ge 0}^{n+g-m}$ where $\Delta$ has $m$ edges, $g$ interior lattice points and $n$ boundary lattice points, solving a conjecture of Crétois and Lang. Additionally, we use abstract tropical curves to construct a compactification of this moduli space by adding points that correspond to collections of curves that can be patchworked together to produce a curve in ${\mathcal{H}}_{\Delta }$. This compactification comes with a natural stratification with the same poset as the secondary polytope of $\Delta$.

Several significant changes have been made to this manuscript since the first time it was uploaded. In particular, in v2 section 5 was added and in v4 section 6 was added. Some of the proofs have been replaced by simpler ones

References in corpus (4)

• Patchworking real algebraic varieties
• Regularity theorem for totally nonnegative flag varieties
• The vanishing cycles of curves in toric surfaces I
• On the curvature of the real amoeba

Cited by in corpus (1)

• Monodromy of rational curves on toric surfaces