This article is the first in a series of two in which we study the vanishing cycles of curves in toric surfaces. We give a list of possible obstructions to contract vanishing cycles within a given complete linear system. Using tropical means, we show that any non-separating simple closed curve is a vanishing cycle whenever none of the listed obstructions appears.

For an ample line bundle L on a complete toric surface X, we consider the subset V L ⊂ |L| of irreducible, nodal, rational curves contained in the smooth locus of X. We study the monodromy map from the fundamental group of V L to the permutation group on the set of nodes of a reference curve C ∈ V L. We identify a certain obstruction map Ψ X defined on the set of nodes of C and show that the image of the monodromy is exactly the group of deck transformations of Ψ X , provided that L is sufficiently big (in the sense we make precise below). Along the way, we construct a handy tool to compute the image of the monodromy for any pair (X, L). Eventually, we present a family of pairs (X, L) with small L and for which the image of the monodromy is strictly smaller than expected.

We resume the study initiated in [CL17]. For a generic curve C in an ample linear system |L| on a toric surface X, a vanishing cycle of C is an isotopy class of simple closed curve that can be contracted to a point along a degeneration of C to a nodal curve in |L|. The obstructions that prevent a simple closed curve in C from being a vanishing cycle are encoded by the adjoint line bundle K X ⊗ L. In this paper, we consider the linear systems carrying the two simplest types of obstruction. Geometrically, these obstructions manifest on C respectively as an hyperelliptic involution and as a spin structure. In both cases, we determine all the vanishing cycles by investigating the associated monodromy maps, whose target space is the mapping class group M CG(C). We show that the image of the monodromy is the subgroup of M CG(C) preserving respectively the hyperelliptic involution and the spin structure. The results obtained here support the Conjecture 1 in [CL17] aiming to describe all the vanishing cycles for any pair (X, L).

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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 → CP 1 relates the latter problem to the study of the Lyashko-Looijenga map ( ). The map associates to a generic curve V ∈ |L| the unordered set of the critical values of γ on CP 1 . We prove two statements concerning that are crucial for our classification problem: the map is algebraic; the map extends to nodal curves in |L|. This fact allows us to construct many examples of pairs (V, S) by perturbing nodal curves. Contents