# Amoebas of curves and the Lyashko-Looijenga map

## TL;DR

This paper investigates the critical locus of the moment map for curves in toric surfaces, using the Lyashko-Looijenga map to classify pairs of curves and their critical loci, revealing algebraic properties and extension to nodal curves.

## Contribution

It introduces the algebraic nature of the Lyashko-Looijenga map and its extension to nodal curves, enabling classification and construction of examples of curve-critical locus pairs.

## Key findings

- Critical locus is smooth except on real codimension 1 walls.
- Lyashko-Looijenga map is algebraic.
- Extension of the map to nodal curves allows for example construction.

## 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 $\gamma : \mathcal{V} \rightarrow \mathbb{C}P^1$. 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.