Related papers: Reconstructing plane quartics from their invariant…
We present some results concerning the Saito module and the torsion submodule of an analytic plane curve, and we provide a method for computing them. Using this algorithm, we compute analytic invariants for plane curves with multiplicity…
In this paper, we present two related results on curves of genus 3. The first gives a bijection between the classes of the following objects: * Smooth non-hyperelliptic curves C of genus 3, with a choice of an element a in Jac(C)[2]-{0},…
We describe some regular techniques of calculating finite degree invariants of triple points free smooth plane curves $S^1 \to R^2$. They are a direct analog of similar techniques for knot invariants and are based on the calculus of {\em…
Eliminating the arbitrary coefficients in the equation of a generic plane curve of order $n$ by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to…
One of the general problems in algebraic geometry is to determine algorithmically whether or not a given geometric object, defined by explicit polynomial equations (e.g. a curve or a surface), satisfies a given property (e.g. has…
In this paper, we discuss some problems of elementary plane differential geometry and kinematics. Although the results are not new, the consistent use of complex-valued functions (plane curves) of a real variable (parameter) allows to…
We compute the Welschinger invariants of blowups of the projective plane at an arbitrary conjugation invariant configuration of points. Specifically, open analogues of the WDVV equation and Kontsevich-Manin axioms lead to a recursive…
We consider a plane polynomial vector field $P(x,y)dx+Q(x,y)dy$ of degree $m>1$. To each algebraic invariant curve of such a field we associate a compact Riemann surface with the meromorphic differential $\omega=dx/P=dy/Q$. The asymptotic…
Given an odd representation of the absolute Galois group of Q onto PGL(2,3) and a positive integer N, there exists a twisted modular curve defined over Q whose rational points classify the quadratic Q-curves of degree N realizing the…
The kernel method is an essential tool for the study of generating series of walks in the quarter plane. This method involves equating to zero a certain polynomial, the kernel polynomial, and using properties of the curve, the kernel curve,…
We present an algorithm for constructing the fixed point of a general non-isometric similarity of the plane.
In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with at least five invariant lines, including the line at…
Motivated by the analogies between the projective and the almost quaternionic geometries, we study the generalized planar curves and mappings. We follow, recover, and extend the classical approach as developed by Mikes and Sinyukov. Then we…
We construct a surface of general type with invariants \( \chi = K^2 = 1 \) and torsion group \( \Bbb{Z}/{2} \). We use a double plane construction by finding a plane curve with certain singularities, resolving these, and taking the double…
The main goal of this article is to expand the theory of invariants of Artin-Schreier curves by giving a complete classification in genus 3 and 4. To achieve this goal, we first establish standard forms of Artin-Schreier curves and…
In the present paper, we revisit the geometry of smooth plane quartics and their bitangents from several perspectives. First, we study in detail the weak combinatorics of arrangements of bitangents associated with highly symmetric quartic…
As another application of the degeneration methods of [V3], we count the number of irreducible degree $d$ geometric genus $g$ plane curves, with fixed multiple points on a conic $E$, not containing $E$, through an appropriate number of…
Reconstructing the missing parts of a curve has been the subject of much computational research, with applications in image inpainting, object synthesis, etc. Different approaches for solving that problem are typically based on processes…
New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all…
We construct special conics configurations from some points configurations which are the singularities of the dual of a quartic curve.