Lionel Lang

• Lecturer at University of Gävle

We establish a patchworking theorem à la Viro for the Log-critical locus of algebraic curves in ( ℂ ∗ ) 2 {(\mathbb{C}^{\ast})^{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 fa...

A generic polynomial f(x,y,z) with a prescribed Newton polytope defines a symmetric spatial curve f(x,y,z)=f(y,x,z)=0. We study its geometry: the number, degree and genus of its irreducible components, the number and type of singularities, etc. and discuss to what extent these results generalize to higher dimension and more complicated symmetries....

We address two interrelated problems concerning permutation of roots of univariate polynomials whose coefficients depend on parameters. First, we compute the Galois group of polynomials $\varphi(x)\in\mathbb{C}[t_1,\cdots,t_k][x]$ over $\mathbb{C}(t_1,\cdots,t_k)$. Provided that the corresponding multivariate polynomial $\varphi(x,t_1,\cdots,t_k)$...

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 w...

We establish a patchworking theorem \`a la Viro for the Log-critical locus of algebraic curves in $(\mathbb{C}^*)^2$. 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...

In this notte, we study the maximal number of intersection points of a line with the contour of a hypersurface amoeba in R^n

In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We generalize the results of \cite{L19, Tyom13, Tyom14}, and 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 except...

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 ca...

Let $\mathcal{C}_d\subset \mathbb{C}^{d+1}$ be the space of non-singular, univariate polynomials of degree $d$. The Vi\`{e}te map $\mathscr{V} : \mathcal{C}_d \rightarrow Sym_d(\mathbb{C})$ sends a polynomial to its unordered set of roots. It is a classical fact that the induced map $\mathscr{V}_*$ at the level of fundamental groups realises an iso...

In this note, we investigate the maximal number of intersection points of a line with the contour of hypersurface amoebas in $\mathbb{R}^n$. We define the latter number to be the $\mathbb{R}$-degree of the contour. We also investigate the $\mathbb{R}$-degree of related sets such as the boundary of amoebas and the amoeba of the real part of hypersur...

For an ample line bundle $\mathcal{L}$ on a complete toric surface $X$, we consider the subset $V_{\mathcal{L}} \subset \vert \mathcal{L} \vert$ of irreducible nodal rational curves contained in the smooth locus of $X$. We study the monodromy map from the fundamental group of $V_{\mathcal{L}}$ to the permutation group on the set of nodes of a refer...

A system of $n$ polynomial equations in $n$ variables with indeterminate coefficients is said to be reduced and irreducible, if it cannot be simplified by monomial changes of variables (see below for a formal definition). It was proved in [E18] that the monodromy of such system is the symmetric group, and conjectured, that the monodromy of a non-re...

We resume the study initiated in \cite{CL}. For a generic curve $C$ in an ample linear system $\vert \mathcal{L} \vert$ 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 $\vert \mathcal{L} \vert$. The obstructions that prevent...

We resume the study initiated in \cite{CL}. For a generic curve $C$ in an ample linear system $\vert \mathcal{L} \vert$ 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 $\vert \mathcal{L} \vert$. The obstructions that prevent...

For any curve $\mathcal{V}$ in a toric surface $X$, we study the critical locus $S(\mathcal{V})$ of the moment map $\mu$ from $\mathcal{V}$ to its compactified amoeba $\mu(\mathcal{V})$. We show that for curves $\mathcal{V}$ in a fixed complete linear system, the critical locus $S(\mathcal{V})$ is smooth apart from some real codimension $1$ walls....

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 obstru...

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 obstru...

Simple Harnack curves, introduced in \cite{Mikh}, are smooth real algebraic curves in maximal position in toric surfaces. In the present paper, we suggest a natural generalization of simple Harnack curves by relaxing the smoothness assumption. After mentioning some of their properties, we address the question of their construction. We define tropic...

Harmonic amoebas are generalizations of amoebas of algebraic curves immersed in complex tori. Introduced in \cite{Kri}, the consideration of such objects suggests to enlarge the scope of classical tropical geometry of curves. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces, and show how they...