arXiv subjects

Lionel Lang

Publications and source records attributed to Lionel Lang.

13 recordsLinked to original sources

arXiv2022

Bernstein-Kouchnirenko-Khovanskii with a symmetry

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. As an application, we characterize generic one-parameter families of complex univariate polynomials, whose Galois group is a complete symmetric group.

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arXiv2022

Permuting the roots of univariate polynomials whose coefficients depend on parameters

We address two interrelated problems concerning the permutation of roots of univariate polynomials whose coefficients depend on parameters. First, we compute the Galois group of polynomials $\varphi(x)\in\mathbb{C}[y_1,\cdots,y_k][x]$ over $\mathbb{C}(y_1,\cdots,y_k)$. Provided that the corresponding multivariate polynomial $\varphi(x,y_1,\ldots,y_k)$ is generic with respect to its support $A\subset \mathbb{Z}^{k+1}$, we determine the associated Galois group for any such $A$. Second, we determine the Galois group of systems of polynomial equations of the form $p(x,y)=q(y)=0$ where $p$ and $q$ have fixed supports $A_1\subset \mathbb{Z}^2$ and $A_2\subset \{0\}\times \mathbb{Z}$, respectively. For each problem, we determine the image of an appropriate braid monodromy map in order to compute the sought Galois group. Among the applications, we determine the Galois group of any rational function generic with respect to its support. We also provide general obstructions to the Galois group of enumerative problems over algebraic groups.

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arXiv2021

Patchworking the Log-critical locus of planar curves

We establish a patchworking theorem à 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 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.

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arXiv2020

Monodromy of rational curves on toric surfaces

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 reference curve $C \in V_{\mathcal{L}}$. We identify a certain obstruction map $\varPsi_{X}$ defined on the set of nodes of $C$ and show that the image of the monodromy is exactly the group of deck transformations of $\varPsi_{X}$, provided that $\mathcal{L}$ is sufficiently big (in a sense we precise below). Along the way, we provide a handy tool to compute the image of the monodromy for any pair $(X, \mathcal{L})$. Eventually, we present a family of pairs $(X, \mathcal{L})$ with small $\mathcal{L}$ and for which the image of the monodromy is strictly smaller than expected.

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arXiv2020

A note on the Severi problem for toric surfaces

In this note, we make a step towards the classification of toric surfaces admitting reducible Severi varieties. We generalize the results of [Lan19, Tyo13, Tyo14], 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 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.

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arXiv2020

Braid monodromy of univariate fewnomials

Let $\mathcal{C}_d\subset \mathbb{C}^{d+1}$ be the space of non-singular, univariate polynomials of degree $d$. The Viète 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 isomorphism between $π_1(\mathcal{C}_d)$ and the Artin braid group $B_d$. For fewnomials, or equivalently for the intersection $\mathcal{C}$ of $\mathcal{C}_d$ with a collection of coordinate hyperplanes in $\mathbb{C}^{d+1}$, the image of the map $\mathscr{V} _* : π_1(\mathcal{C}) \rightarrow B_d$ is not known in general. In the present paper, we show that the map $\mathscr{V} _*$ is surjective provided that the support of the corresponding polynomials spans $\mathbb{Z}$ as an affine lattice. If the support spans a strict sublattice of index $b$, we show that the image of $\mathscr{V} _*$ is the expected wreath product of $\mathbb{Z}/b\mathbb{Z}$ with $B_{d/b}$. 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.

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arXiv2020

Sparse polynomial equations and other enumerative problems whose Galois groups are wreath products

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 $f(x) = c_0 + c_1 x^{d_1} + \ldots + c_k x^{d_k}$ by varying its coefficients. If the GCD of the exponents is $d$, then the polynomial admits the change of variable $y=x^d$, 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 $d_k/d$ elements and $\mathbb{Z}/d\mathbb{Z}$. The aim of this paper is to prove this equality and study its multidimensional generalization: we show that 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.

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arXiv2020

Harmonic tropical morphisms and approximation

\textit{Harmonic amoebas} are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced in \cite{Kri}, 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. in \cite{Mikh06}. 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.

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arXiv2019

A generalisation of simple Harnack curves

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 $(\mathbb{C}^*)^2$ with totally real logarithmic Gauss map. We investigate which of the many properties of simple Harnack curves survive the latter generalisation. We also show how tropical geometry allows to construct plenty of examples. Since generalised Harnack curves can develop arbitrary singularities, in contrast with the original definition where only real isolated double points can appear, we pay a special attention to the simplest new instance of generalised Harnack curves, namely curves with a single hyperbolic node. In particular, we give their topological classification as in \cite{Mikh} and show how such curves can be recovered from their spine.

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arXiv2019

On the number of intersection points of the contour of an amoeba with a line

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 hypersurfaces defined over $\mathbb{R}$. For all these objects, we provide bounds for the respective $\mathbb{R}$-degrees.

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arXiv2019

The vanishing cycles of curves in toric surfaces II

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 a simple closed curve in $C$ from being a vanishing cycle are encoded by the adjoint line bundle $K_X \otimes \mathcal{L}$. 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 $MCG(C)$. We show that the image of the monodromy is the subgroup of $MCG(C)$ preserving respectively the hyperelliptic involution and the Spin structure. In particular, we provide an explicit finite set of generators for the Spin mapping class group. The results obtained here support the Conjecture $1$ in \cite{CL} aiming to describe all the vanishing cycles for any pair $(X, \mathcal{L})$.

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arXiv2017

Amoebas of curves and the Lyashko-Looijenga map

For any curve $\mathcal{V}$ in a toric surface $X$, we study the critical locus $S(\mathcal{V})$ of the moment map $μ$ from $\mathcal{V}$ to its compactified amoeba $μ(\mathcal{V})$. 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. We then investigate the topological classification of pairs $(\mathcal{V},S(\mathcal{V}))$ when $\mathcal{V}$ and $S(\mathcal{V})$ are smooth. As a main tool, we use the Lyashko-Looijenga mapping ($\mathcal{LL}$) relative to the logarithmic Gauss map $γ: \mathcal{V} \rightarrow \mathbb{C}P^1$. We prove two statements concerning $\mathcal{LL}$ that are crucial for our study: the map $\mathcal{LL}$ is algebraic; the map $\mathcal{LL}$ extends to nodal curves. It allows us to construct many examples of pairs $(\mathcal{V},S(\mathcal{V}))$ by perturbing nodal curves.

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arXiv2017

The vanishing cycles of curves in toric surfaces I

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.

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