Amoebas of curves and the Lyashko-Looijenga map

Abstract

For any curve $\mathcal{V}$ in a toric surface $X$, we study the critical locus $S(\mathcal{V})$ of the moment map $\mu$ from $\mathcal{V}$ to its compactified amoeba $\mu (\mathcal{V})$. We show that for curves $\mathcal{V}$ in a fixed complete linear system, the critical locus $S(\mathcal{V})$ is smooth apart from some real codimension $1$ walls. We then investigate the topological classification of pairs $(\mathcal{V},S(\mathcal{V}))$ when $\mathcal{V}$ and $S(\mathcal{V})$ are smooth. As a main tool, we use the Lyashko-Looijenga mapping ($\mathcal{LL}$) relative to the logarithmic Gauss map $\gamma :\mathcal{V}\to \mathbb{C}{P}^1$. We prove two statements concerning $\mathcal{LL}$ that are crucial for our study: the map $\mathcal{LL}$ is algebraic; the map $\mathcal{LL}$ extends to nodal curves. It allows us to construct many examples of pairs $(\mathcal{V},S(\mathcal{V}))$ by perturbing nodal curves.

25 pages, 4 figures

References in corpus (2)

• Existence of unimodular triangulations - positive results
• Harmonic tropical morphisms and approximation

Cited by in corpus (1)

• On the number of intersection points of the contour of an amoeba with a line