Amoebas of curves and the Lyashko–Looijenga map
Lionel Lang · Journal of the London Mathematical Society · 2019
For any curve V in a toric surface X, we study the critical locus S ⊂ V of the moment map μ from V to its compactified amoeba μ ( V ) . For any complete linear system | L | given by an ample line bundle L on X, we show that the critical locus S ⊂ V is smooth as long as the curve V is outside of a subset of real codimension 1 in | L | . In particular, the complement of the latter subset appears to be disconnected for general L. It suggests a classification problem analogous to Hilbert's Sixteenth Problem, namely the topological classification of pairs ( V , S ) for curves V ∈ | L | . The description of the critical locus S in terms of the logarithmic Gauß map γ : V → C P 1 relates the latter problem to the study of the Lyashko–Looijenga map ( ℓ ℓ ). The map ℓ ℓ associates to a generic curve V ∈ | L | the unordered set of the critical values of γ on C P 1 . We prove two statements concerning ℓ ℓ that are crucial for our classification problem: the map ℓ ℓ is algebraic; the map ℓ ℓ extends to nodal curves in | L | . This fact allows us to construct many examples of pairs ( V , S ) by perturbing nodal curves.