arXiv research

Lie algebroids and curved Lie algebras are equivalent categories.

Research examines curves of degree 8 with specific singularities.

Recalls and refines the concept of algebraically rectifiable curves.

The paper describes handle decompositions and Kirby diagrams for plane algebraic curves.

Study on algebraic curves' invariants and vanishing criteria.

Analytic curves linked to algebraic ones via Schottky groups.

The paper studies spaces of non-compact real algebraic curves and their uniformisation.

The coamoeba of any complex algebraic plane curve $V$ is its image in the real torus under the argument map. The area counted with multiplicity of the coamoeba of any algebraic curve in $(\mathbb{C}^*)^2$ is bounded in terms of the degree of the curve. We show in this Note that up to multiplication by a constant in $(\…

The study refines algebraic domains with specific boundary conditions.

We give some new congruences for singular real algebraic curves which generalize Fiedler's congruence for nonsingular curves.

We define a (co-)Poisson (co)algebra of curves on a bordered surface. A bordered surface is a surface whose boundary have marked points. Curves on the bordered surface are oriented loops and oriented arcs whose endpoints in the set of marked points. We define a (co-)Poisson (co)bracket on the symmetric algebra of a quo…

Unique Teichmüller curve found in complex geometry.

New insights into algebraic geometry of a conjecture, leading to origami curves.

Invariant structures link to algebraic curves with specific properties.

A Teichmüller curve is an algebraic and isometric immersion of an algebraic curve into the moduli space of Riemann surfaces. We give the first explicit algebraic models of Teichmüller curves of positive genus. Our methods are based on the study of certain Hilbert modular forms and the use of Ahlfors's variational formu…

The paper calculates period matrices for specific algebraic curves.

Algorithm constructs algebraic curves from translation surfaces.

Researchers describe a Ceresa class for tropical and topological curves, linking algebraic and cohomological perspectives.

New examples of translation surfaces on hyperelliptic curves with many automorphisms.

The paper extends a geometric model using singular curves.

The abstract aims to generalize classical curve concepts to uniquely define complex curves.

Research on refined algebraic domains respecting differential geometry.

A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus $g\ge 1$ is itself a ruled surface over a curve of genus $g$. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in case $g\ge 2$.

T-curves are piecewise linear curves which have been used with success since the beginning of the 1990's to construct new real algebraic curves with prescribed topology mainly on the real projective plane. In fact T-curves can be used on any real projective toric surface. We generalize here the construction of the latt…

A Lie algebra structure on variation vector fields along an immersed curve in a $2$-dimensional real space form is investigated. This Lie algebra particularized to plane curves is the cornerstone in order to define a Hamiltonian structure for plane curve motions. The Hamiltonian form and the integrability of the planar…

This paper is motivated by the real symplectic isotopy problem : does there exists a nonsingular real pseudoholomorphic curve not isotopic in the projective plane to any real algebraic curve of the same degree? Here, we focus our study on symmetric real curves on the projective plane. We give a classification of real s…

I construct "fake algebraic curves" in $Cp^2$. More precisely, for any k>2, I construct infinitely many pairwise smoothly non-isotopic (and moreover not ambient diffeomorphic) smooth surfaces $F\subset Cp^2$ homeomorphic to a non-singular algebraic curve of degree 2k, realizing the same homology class as such a curve a…

A collection $Δ$ of simple closed curves on an orientable surface is an algebraic $k$-system if the algebraic intersection number $\langle α,β\rangle$ is equal to $k$ in absolute value for every $α, β\in Δ$ distinct. Generalizing a theorem of [MRT14] we compute that the maximum size of an algebraic $k$-system of c…

In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which have intersections with the original curves and surfaces at the singular points. …

Researchers reconstruct algebraic maps onto curves based on prescribed Reeb graphs.

We use Morse theoretical arguments to study algebraic curves in C^2. We take an algebraic curve C in C^2 and intersect it with a family of spheres with fixed origin and varying radii. We explain in detail how does the resulting link change when we cross a singular point of C. Applying link invariants as Murasugi's sign…

In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) …

Classifies semi-algebraic surfaces up to bi-Lipschitz homeomorphisms.

Study on complex curves in hypercomplex nilmanifolds with quaternionic-solvable Lie algebras.

The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves in real toric surfaces is a classical subject that has undergone considerable evolution. On the other hand, not much is …

Conditions for curves on a torus with specific pairwise intersections.

For a non-singular real algebraic projective curve, topological restrictions on a closed motion of a simple real divisor in its linear equivalence class are found.

We prove that every algebraic curve X defined over the algebraic closure of the rationals is birational over the complex numbers to a Teichmuller curve.

Study algebraic curves in C^2 using Floer theory.

We describe two constructions giving rise to curved $A_{\infty}$-algebras. The first consists of deforming $A_{\infty}$-algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures along chain contractions. As an application of the second construction, given a …

New invariant links graph structure to tropical curve properties.

Criteria for extending degree-2 Azumaya algebras with C2-actions over curves.

Alternative proof for non-existence of complete curves in differential strata.

Researchers compute c-projective symmetry algebras for Kähler surfaces.

Let $N_g^k$ be a nonorientable surface of genus \ $g\geq 5$ \ with \ $k$-punctures. In this note, we will give an algebraic characterization of a Dehn twist about a simple closed curve on $N_g^k$. Along the way, we will fill some little gaps in the proofs of some theorems in \cite{A} and \cite{I1} giving algebraic char…

The study connects curves and cohomology on manifolds.

Constructs Lagrangian skeleta for curve singularities.

We complete the topological classification of real algebraic non-singular curves of bidegree $(5, 5)$ on the quadric ellipsoid. We show in particular that previously known restrictions form a complete system for this bidegree. Therefore, the main part of the paper concerns the construction of real algebraic curves. Our…