Singular Curves Research Articles

Polar germs, Jacobian ideal and analytic classification of irreducible plane curve singularities

We examine the relationship between the analytic type of an irreducible plane curve singularity with a single characteristic exponent and the germs of curve defined by the elements of its Jacobian ideal, in particular its polar germs.

Automorphisms of tropical Hassett spaces

Given an integer g \geq 0 and a weight vector w \in \mathbb{Q}^n \cap (0, 1]^n satisfying 2g - 2 +\sum w_i > 0 , let \Delta_{g, w} denote the moduli space of n -marked, w -stable tropical curves of genus g and volume one. We calculate the automorphism group \operatorname{Aut}(\Delta_{g, w}) for g \geq 1 and arbitrary w , and we calculate the group \operatorname{Aut}(\Delta_{0, w}) when w is heavy/light. In both of these cases, we show that \operatorname{Aut}(\Delta_{g, w}) \cong \operatorname{Aut}(K_w) , where K_w is the abstract simplicial complex on \{1, \ldots, n\} whose faces are subsets with w -weight at most 1. We show that these groups are precisely the finite direct products of symmetric groups. The space \Delta_{g, w} may also be identified with the dual complex of the divisor of singular curves in the algebraic Hassett space \overline{\mathcal{M}}_{g, w} . Following the work of Massarenti and Mella (2017) on the biregular automorphism group \operatorname{Aut}(\overline{\mathcal{M}}_{g, w}) , we show that \operatorname{Aut}(\Delta_{g, w}) is naturally identified with the subgroup of automorphisms which preserve the divisor of singular curves.

Electromagnetic curves and Berry phase construction of a polarized light wave along an optical fiber which is a singular curve on [formula omitted

In this study, we investigate the cases of a linearly polarized light wave along an optical fiber which is accepted as a singular curve on S2 and the rotation of the polarization planes. Also, we construct a Berry-phase model of polarized light wave along an optical fiber with singular points. In addition, we introduce two different Rytov curves according to the polarization vector E being perpendicular to the defined planes along an optical fiber with singular points. Finally, we define the Lorentz force equations for all cases and we examine the electromagnetic trajectories created by the electric field of the light wave traveling in the singular optical fiber.

Representations of the fundamental group and Higgs bundles on singular integral curves

Representations of the fundamental group and Higgs bundles on singular integral curves

Four-Dimensional Gait Surfaces for a Tilt-Rotor—Two Color Map Theorem

This article presents the four-dimensional surfaces that guide the gait plan for a tilt-rotor. The previous gaits analyzed in the tilt-rotor research are inspired by animals; no theoretical base backs the robustness of these gaits. This research deduces the gaits by diminishing the adverse effect of the attitude of the tilt-rotor for the first time. Four-dimensional gait surfaces are subsequently found on which the gaits are expected to be robust to the attitude. These surfaces provide the region where the gait is suggested to be planned. However, a discontinuous region may hinder the gait plan process while utilizing the proposed gait surfaces. The ‘Two Color Map Theorem’ is then established to guarantee the continuity of each gait designed. The robustness of the typical gaits on the gait surface, obeying the Two Color Map Theorem, is demonstrated by comparing the singular curves in attitude with the gaits not on the gait surface. The result shows that the gaits on the gait surface receive wider regions of the acceptable attitudes.

Morsifications and mutations

We describe and investigate a connection between the topology of isolated singularities of plane curves and the mutation equivalence, in the sense of cluster algebra theory, of the quivers associated with their morsifications.

Bertrand and Mannheim Curves of Spherical Framed Curves in a Three-Dimensional Sphere

We investigated differential geometries of Bertrand curves and Mannheim curves in a three-dimensional sphere. We clarify the conditions for regular spherical curves to become Bertrand and Mannheim curves. Then, we concentrate on Bertrand and Mannheim curves of singular spherical curves. As singular spherical curves, we considered spherical framed curves. We define Bertrand and Mannheim curves of spherical framed curves. We give conditions for spherical framed curves to become Bertrand and Mannheim curves.

Lagrangian skeleta and plane curve singularities

We construct closed arboreal Lagrangian skeleta associated to links of isolated plane curve singularities. This yields closed Lagrangian skeleta for Weinstein pairs (mathbb {C}^2,Lambda ) and Weinstein 4-manifolds W(Lambda ) associated to max-tb Legendrian representatives of algebraic links Lambda subseteq (mathbb {S}^3,xi _text {st}). We provide computations of Legendrian and Weinstein invariants, and discuss the contact topological nature of the Fomin–Pylyavskyy–Shustin–Thurston cluster algebra associated to a singularity. Finally, we present a conjectural ADE-classification for Lagrangian fillings of certain Legendrian links and list some related problems.

Strong Sard conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3

In this paper we prove the strong Sard conjecture for sub-Riemannian structures on 3-dimensional analytic manifolds. More precisely, given a totally nonholonomic analytic distribution of rank 2 on a 3-dimensional analytic manifold, we investigate the size of the set of points that can be reached by singular horizontal paths starting from a given point and prove that it has Hausdorff dimension at most 1. In fact, provided that the lengths of the singular curves under consideration are bounded with respect to a given complete Riemannian metric, we demonstrate that such a set is a semianalytic curve. As a consequence, combining our techniques with recent developments on the regularity of sub-Riemannian minimizing geodesics, we prove that minimizing sub-Riemannian geodesics in 3-dimensional analytic manifolds are always of class C^1, and actually they are analytic outside of a finite set of points.

On correctness of conditions for the CSIDH algorithm implementation on Edwards curves

Incorrect formulation and incorrect solution of the problem of the CSIDH algorithm implementation on Edwards curves was revealed in one of the famous works. The purpose of this paper is to present a detailed critique of such concept with a proof of its inconsistency. Specific properties of three non-isomorphic classes of super singular curves in the generalized Edwards form are considered: full, quadratic, and twisted Edwards curves. Conditions for existence of curves of all 3 classes with order of curves over a prime field are determined. The implementation of the CSIDH algorithm on isogenies of odd prime degrees is based on the use of quadratic twist pairs of elliptic curves. To this end, the CSIDH algorithm can be built both on complete Edwards curves with quadratic twist within this class, and on quadratic and twisted Edwards curves forming pairs of quadratic twist. In contrast to this, the authors of a well-known work are trying to prove theorems that state that there is a solution within one class of curves with a parameter which is a square. The critical analysis of theorems, lemmas, erroneous statements in this work is carried out. Theorem 2 on quadratic twist in classes of Edwards curves is proved. The CSIDH algorithm modification based on isogenies of quadratic and twisted Edwards curves is presented. To illustrate the correct solution of the problem, an example of Alice and Bob calculations in the secret sharing scheme according to the CSIDH algorithm is considered for.

MULTI VARIATE NEURO-STATISTICAL SPARSE TRANSFORM FOR GRAY SCALE IMAGES

Abstract-- The main objective of this paper is to examine the performance of Neuro- Statistical sparse transformation function for implementation in a still image vector coding based compression system. This paper discusses the important features of low bit-rate image coding which is based on recent developments in the theory of multivariate nonlinear piecewise polynomial approximation in still images. It combines Binary Space Partition (BSP) scheme with Geometric Wavelet (GW) tree approximation so as to efficiently capture curve singularities and provide a sparse representation of the image. The quality of the reconstructed image is measured objectively using Peak Signal to Noise Ratio. Experimental results show that the proposed image compression system yields higher compression with minimal loss.

Generalized Affine Springer Theory and Hilbert Schemes on Planar Curves

AbstractWe show that Hilbert schemes of planar curve singularities and their parabolic variants can be interpreted as certain generalized affine Springer fibers for $GL_n$, as defined by Goresky–Kottwitz–MacPherson. Using a generalization of affine Springer theory for Braverman–Finkelberg–Nakajima’s Coulomb branch algebras, we construct a rational Cherednik algebra action on the homology of the Hilbert schemes and compute it in examples. Along the way, we generalize to the parahoric setting the recent construction of Hilburn–Kamnitzer–Weekes, which may be of independent interest. In the spherical case, we make our computations explicit through a new general localization formula for Coulomb branches. Via results of Hogancamp–Mellit, we also show the rational Cherednik algebra acts on the HOMFLY-PT homologies of torus knots. This work was inspired in part by a construction in 3D ${\mathcal {N}}=4$ gauge theory.

On a quadratic form associated with a surface automorphism and its applications to Singularity Theory

We study the nilpotent part N′ of a pseudo-periodic automorphism h of a real oriented surface with boundary Σ. We associate a quadratic form Q defined on the first homology group (relative to the boundary) of the surface Σ. Using the twist formula and techniques from mapping class group theory, we prove that the form Q̃ obtained after killing kerN is positive definite if all the screw numbers associated with certain orbits of annuli are positive. We also prove that the restriction of Q̃ to the absolute homology group of Σ is even whenever the quotient of the Nielsen–Thurston graph under the action of the automorphism is a tree. The case of monodromy automorphisms of Milnor fibers Σ=F of germs of curves on normal surface singularities is discussed in detail, and the aforementioned results are specialized to such situation. This form Q is determined by the Seifert form but can be much more easily computed. Moreover, the form Q̃ is computable in terms of the dual resolution or semistable reduction graph, as illustrated with several examples. Numerical invariants associated with Q̃ are able to distinguish plane curve singularities with different topological types but same spectral pairs. Finally, we discuss a generic linear germ defined on a superisolated surface. In this case the plumbing graph is not a tree and the restriction of Q̃ to the absolute monodromy of Σ=F is not even.

A Compactification of the Universal Moduli Space of Principal G-Bundles

Let G be a semisimple linear algebraic group, rho :Ghookrightarrow text {SL}(V) a finite-dimensional faithful representation, gge 2 a natural number, and delta a positive rational number. We prove the existence of a compactification of the universal moduli space of semistable principal G-bundles over overline{text {M}}_{g}, provided that delta is sufficiently large, having the following property: the fibers over singular curves are the moduli spaces of delta -semistable singular principal G-bundles.

Two algorithms for computing the general component of jet scheme and applications

Let k be a perfect field. Let X be an integral k-variety. Let m∈N. In this article, we study, from the theoretical and computational points of view, the component Gm(X) of the jet scheme Lm(X) defined to be the Zariski closure of the set of truncated arcs with a regular base-point. We prove that Gm(X) can be described from any smooth birational model of X. When X is supposed to be affine embedded in AkN, this description allows us to provide an algorithm, valid in arbitrary characteristic, which computes a Gröbner basis of a presentation of Gm(X) in Ak(m+1)N from the datum of a given explicit smooth affine birational model of X. Then, we show how to use the datum of a presentation of Gm(X) (in the suitable affine space) to deduce algebraic and geometric properties in two contexts from the theoretical and computational points of view. Firstly, we apply our main result to obtain bases for a class of homogeneous differential operators logarithmic along plane curves in arbitrary characteristics. Secondly, we introduce a motivic power series which encodes the geometry of all the Gm(X) and prove its rationality in the specific case of homogeneous plane curve singularities thanks to our description of Gm(X)via the normalization of X.

Gröbner cells of punctual Hilbert schemes in dimension two

We begin with a comprehensive discussion of the punctual Hilbert scheme of the regular two-dimensional local ring in terms of the Gröbner cells. These schemes are the most degenerate fibers of the Grothendieck-Deligne norm map (the Hilbert-Chow morphism), playing an important role in the study of Hilbert schemes of smooth surfaces. They are generally singular, but their Gröbner cells are affine spaces; they admit an explicit parametrization due to Conca and Valla. We use this to obtain the Gröbner decomposition of compactified Jacobians of plane curve singularities, which is non-trivial even for the generalized Jacobians (principal ideals only). One of the applications is the topological invariance of certain variants of compactified Jacobians and the corresponding motivic superpolynomials for analytic deformations of quasi-homogeneous plane curve singularities and some similar families.

On the Bounded Negativity Conjecture and Singular Plane Curves

There are no known failures of Bounded Negativity in characteristic 0. In the light of recent work showing the Bounded Negativity Conjecture fails in positive characteristics for rational surfaces, we propose new characteristic free conjectures as a replacement. We also develop bounds on numerical characteristics of curves constraining their negativity. For example, we show that the $H$-constant of a rational curve $C$ with at most $9$ singular points satisfies $H(C)>-2$ regardless of the characteristic.

On the fifth Whitney cone of a complex analytic curve

We present an explicit procedure to determine the $C_5$-cone of a reduced complex analytic curve $X \subset \mathbb{C}^n$ at a singular point $0 \in X$, which is known to be a union of a finite number of planes. We also give upper bounds on the number of these planes. The procedure comes out with a collections of integers that we call auxiliary multiplicities and we prove they determine the Lipschitz type of complex curve singularities. We finish presenting an example which shows that the number of irreducible components of the $C_5$-cone of a curve is not a bi-Lipschitz invariant.

Convergence of Narasimhan–Simha measures on degenerating families of Riemann surfaces

Given a compact Riemann surface $Y$ and a positive integer $m$, Narasimhan and Simha defined a measure on $Y$ associated to the $m$-th tensor power of the canonical line bundle. We study the limit of this measure on holomorphic families of Riemann surfaces with semistable reduction. The convergence takes place on a hybrid space whose central fiber is the associated metrized curve complex in the sense of Amini and Baker. We also study the limit of the measure induced by the Hermitian pairing defined by the Narasimhan-Simha measure. For $m = 1$, both these measures coincide with the Bergman measure on $Y$. We also extend the definition of the Narasimhan-Simha measure to the singular curves on the boundary of $\overline{\mathcal{M}_g}$ in such a way that these measures form a continuous family of measures on the universal curve over $\overline{\mathcal{M}_g}$.

Plateau’s problem for singular curves

Plateau’s problem for singular curves