The Moduli Space of Harnack Curves in Toric Surfaces

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 \(\mathbb{R}^{m-3}\times\mathbb{R}_{\geq0}^{n+g-m}\) 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 \(\Delta\).