In this paper, we suggest the following generalisation of Mikhalkin’s simple Harnack curves: a generalised simple Harnack curve is a parametrised real algebraic curve in (ℂ^*)2 with totally real logarithmic Gauss map. First, we investigate which of the many properties of simple Harnack curves survive this generalisation. Then, we construct new examples using tropical geometry. Eventually, since generalised Harnack curves can develop arbitrary singularities, in contrast with the original definition, we pay a special attention to the simplest new instance of generalised Harnack curves, namely curves with a single hyperbolic node. In particular, we determine the topological classification of such curves for any given degree.

In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We provide two families of toric surfaces admitting reducible Severi varieties. The first family is general, and is obtained by a quotient construction. The second family is exceptional, and corresponds to certain narrow polygons, which we call kites. We introduce two types of invariants that distinguish between the components of the Severi varieties, and allow us to provide lower bounds on the numbers of the components. The sharpness of the bounds is verified in some cases, and is expected to hold in general for ample enough linear systems. In the appendix, we establish a connection between the Severi problem and the topological classification of univariate polynomials.

We establish a patchworking theorem à la Viro for the Log-critical locus of algebraic curves in (C∗)2(ℂ∗)2 . As an application, we prove the existence of projective curves of arbitrary degree with smooth connected Log-critical locus. To prove our patchworking theorem, we study the behaviour of Log-inflection points along families of curves defined by Viro polynomials. In particular, we prove a generalisation of a theorem of Mikhalkin and the second author on the tropical limit of Log-inflection points.

We introduce a new technique to prove connectivity of subsets of covering spaces (so called inductive connectivity), and apply it to Galois theory of problems of enumerative geometry. As a model example, consider the problem of permuting the roots of a complex polynomial f(x)=c0+c1xd1+⋯+ckxdkf(x)=c0+c1xd1+⋯+ckxdk by varying its coefficients. If the GCD of the exponents is d, then the polynomial admits the change of variable y=xdy=xd, and its roots split into necklaces of length d. At best we can expect to permute these necklaces, i.e. the Galois group of f equals the wreath product of the symmetric group over dk/ddk/d elements and Z/dZZ/dZ. We study the multidimensional generalization of this equality: the Galois group of a general system of polynomial equations equals the expected wreath product for a large class of systems, but in general this expected equality fails, making the problem of describing such Galois groups unexpectedly rich.

Let Cd⊂Cd+1 be the space of nonsingular, univariate polynomials of degree d. The Viète map V:Cd→Symd(C) sends a polynomial to its unordered set of roots. It is a classical fact that the induced map V∗ at the level of fundamental groups realises an isomorphism between π1(Cd) and the Artin braid group Bd. For fewnomials, or equivalently for the intersection C of Cd with a collection of coordinate hyperplanes in Cd+1, the image of the map V∗:π1(C)→Bd is not known in general.

We show that the map V∗ is surjective provided that the support of the corresponding polynomials spans Z as an affine lattice. If the support spans a strict sublattice of index b, we show that the image of V∗ is the expected wreath product of Z∕bZ with Bd∕b. From these results, we derive an application to the computation of the braid monodromy for collections of univariate polynomials depending on a common set of parameters.

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 CC in an ample linear system |L||ℒ| on a toric surface XX, a vanishing cycle of CC is an isotopy class of simple closed curve that can be contracted to a point along a degeneration of CC to a nodal curve in |L||ℒ|. The obstructions that prevent a simple closed curve in CC from being a vanishing cycle are encoded by the adjoint line bundle KX⊗LKX⊗ℒ. In this paper, we consider the linear systems carrying the two simplest types of obstruction. Geometrically, these obstructions manifest on CC 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)MCG(C). We show that the image of the monodromy is the subgroup of MCG(C)MCG(C) preserving respectively the hyperelliptic involution and the spin structure. The results obtained here support Conjecture 11 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)(X,ℒ).