In this note, we investigate the maximal number of intersection points of a line with the contour of a hypersurface amoeba in R-n. We define the latter number to be the R-degree of the contour. We also investigate the R-degree of related sets such as the boundary of an amoeba and the amoeba of the real part of a hypersurface defined over R. For all these objects, we provide bounds for the respective R-degrees.

Harmonic amoebas are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced by Krichever in 2014, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces and show how these morphisms can be systematically described as limits of families of harmonic amoeba maps on Riemann surfaces. It extends previous results about approximation of tropical curves in affine spaces and provides a different point of view on Mikhalkin's approximation Theorem for regular phase-tropical morphisms, as stated e.g. by Mikhalkin in 2006. The results presented here follow from the study of imaginary normalised differentials on families of punctured Riemann surfaces and suggest interesting connections with compactifications of moduli spaces.

For an ample line bundle L on a complete toric surface X, we consider the subset VL subset of|L| of irreducible, nodal, rational curves contained in the smooth locus of X. We study the monodromy map from the fundamental group of VL to the permutation group on the set of nodes of a reference curve C is an element of VL. We identify a certain obstruction map psi 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 psi 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.

For any curve V in a toric surface X, we study the critical locus S subset of V of the moment map mu from V to its compactified amoeba mu(V). For any complete linear system |L| given by an ample line bundle L on X, we show that the critical locus S subset of 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 is an element of|L|. The description of the critical locus S in terms of the logarithmic Gau ss map gamma:V -> CP1 relates the latter problem to the study of the Lyashko-Looijenga map (ll). The map ll associates to a generic curve V is an element of|L| the unordered set of the critical values of gamma on CP1. We prove two statements concerning ll that are crucial for our classification problem: the map ll is algebraic; the map ll extends to nodal curves in |L|. This fact allows us to construct many examples of pairs (V,S) by perturbing nodal curves.

We resume the study initiated in [R. Crétois and L. Lang, The vanishing cycles of curves in toric surfaces, I, preprint (2017), arXiv:1701.00608]. For a generic curve C" role="presentation">C in an ample linear system |ℒ|" role="presentation">|L| on a toric surface X" role="presentation">X, a vanishing cycle of C" role="presentation">C is an isotopy class of simple closed curve that can be contracted to a point along a degeneration of C" role="presentation">C to a nodal curve in |ℒ|" role="presentation">|L|. The obstructions that prevent a simple closed curve in C" role="presentation">C from being a vanishing cycle are encoded by the adjoint line bundle KX⊗ℒ" role="presentation">KX⊗L. In this paper, we consider the linear systems carrying the two simplest types of obstruction. Geometrically, these obstructions manifest on C" role="presentation">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 MCG(C)" role="presentation">MCG(C). We show that the image of the monodromy is the subgroup of MCG(C)" role="presentation">MCG(C) preserving respectively the hyperelliptic involution and the spin structure. The results obtained here support Conjecture 1" role="presentation">1 in [R. Crétois and L. Lang, The vanishing cycles of curves in toric surfaces, I, preprint (2017), arXiv:1701.00608] aiming to describe all the vanishing cycles for any pair (X,ℒ)" role="presentation">(X,L).

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.