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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  1relates 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.

Amoebas of curves and the Lyashko–Looijenga map