Plane Curves With Nodes

1989 ◽  
Vol 41 (2) ◽  
pp. 193-212 ◽  
Author(s):  
Robert Treger
Keyword(s):  
General Position ◽  
Algebraic Curve ◽  
Plane Curve ◽  
Exceptional Case ◽  
Plane Curves ◽  
Nodal Plane ◽  
Nodal Curve

A smooth algebraic curve is birationally equivalent to a nodal plane curve. One of the main problems in the theory of plane curves is to describe the situation of nodes of an irreducible nodal plane curve (see [16, Art. 45], [10], [7, Book IV, Chapter I, §5], [12, p. 584], and [3]).Let n denote the degree of a nodal curve and d the number of nodes. The case (AZ, d) — (6,9) has been analyzed by Halphen [10]. It follows from Lemma 3.5 and Proposition 3.6 that this is an exceptional case. The case d ≦n(n + 3)/6, d ≦(n — 1)(n — 2)/2, and (n, d) ≠ (6,9) was investigated by Arbarello and Cornalba [3]. We present a simpler proof (Corollary 3.8).We consider the main case which is particularly important due to its applications to the moduli variety of curves, compare [19, Chapter VIII, Section 4]. Let Vn,d be the variety of irreducible curves of degree n with d nodes and no other singularities such that each curve of Vn,d can be degenerated into n lines in general position (see [17]).

Design of Mechanisms to Trace Plane Curves

2016 ◽  
Author(s):  
Yang Liu ◽  
J. Michael McCarthy
Keyword(s):  
Fourier Series ◽  
Algebraic Curve ◽  
Design Methodology ◽  
Plane Curve ◽  
Plane Curves ◽  
The Third ◽  
Universality Theorem ◽  
Serial Chain ◽  
Plane Algebraic Curve ◽  
Trace Curves

This paper describes a mechanism design methodology that assembles standard components to trace plane curves that have a Fourier series parameterization. This approach can be used to approximate complex plane curves to interpolate image boundaries constructed from points. We describe three ways to construct a mechanism that generates a curve from a Fourier series parameterization. One uses Scotch yoke linkages for each term of Fourier series which are added using a belt drive. The second approach uses a coupled serial chain for each coordinate Fourier parameterization. The third method uses one constrained coupled serial chain to trace a specified plane curve. This work can be viewed as a version of the Kempe Universality Theorem that states that a linkage exists that can trace any plane algebraic curve. In our case, we include belts and pulleys, and obtain linkages that trace curves that have Fourier parameterizations.

Legitimacy

2019 ◽  
Keyword(s):  
International Law ◽  
The State ◽  
State Sovereignty ◽  
International Courts ◽  
The Rule Of Law ◽  
State Legitimacy ◽  
The Relationship ◽  
Supranational Institutions

This collection brings together scholars of jurisprudence and political theory to probe the question of ‘legitimacy’. It offers discussions that interrogate the nature of legitimacy, how legitimacy is intertwined with notions of statehood, and how legitimacy reaches beyond the state into supranational institutions and international law. Chapter I considers benefit-based, merit-based, and will-based theories of state legitimacy. Chapter II examines the relationship between expertise and legitimate political authority. Chapter III attempts to make sense of John Rawls’s account of legitimacy in his later work. Chapter IV observes that state sovereignty persists, since no alternative is available, and that the success of the assortment of international organizations that challenge state sovereignty depends on their ability to attract loyalty. Chapter V argues that, to be complete, an account of a state’s legitimacy must evaluate not only its powers and its institutions, but also its officials. Chapter VI covers the rule of law and state legitimacy. Chapter VII considers the legitimation of the nation state in a post-national world. Chapter VIII contends that legitimacy beyond the state should be understood as a subject-conferred attribute of specific norms that generates no more than a duty to respect those norms. Chapter IX is a reply to critics of attempts to ground the legitimacy of suprastate institutions in constitutionalism. Chapter X examines Joseph Raz’s perfectionist liberalism. Chapter XI attempts to bring some order to debates about the legitimacy of international courts.

scholarly journals Connected Numbers and the Embedded Topology of Plane Curves

2018 ◽  
Vol 61 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Taketo Shirane
Keyword(s):  
Plane Curve ◽  
Smooth Curve ◽  
Plane Curves ◽  
Irreducible Curve ◽  
Galois Cover ◽  
Splitting Number

AbstractThe splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ 4, where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total inflectional tangents.

scholarly journals Plane Curves Which are Quantum Homogeneous Spaces

2021 ◽  
Author(s):  
Ken Brown ◽  
Angela Ankomaah Tabiri
Keyword(s):  
Hopf Algebras ◽  
Final Section ◽  
Homogeneous Spaces ◽  
Plane Curve ◽  
Plane Curves ◽  
Closed Field ◽  
Quantum Homogeneous Space ◽  
Central Elements ◽  
Pointed Hopf Algebras ◽  
Future Work

AbstractLet $\mathcal {C}$ C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, $\mathcal {C}$ C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring $\mathcal {O}(\mathcal {C})$ O ( C ) of $\mathcal {C}$ C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.

scholarly journals Isotopies of generic plane curves

1983 ◽  
Vol 24 (2) ◽  
pp. 195-206 ◽  
Author(s):  
J. W. Bruce
Keyword(s):  
Plane Curve ◽  
Gauss Map ◽  
Plane Curves ◽  
Bifurcation Set

The aim of this paper is to explore some facets of the geometry of generic isotopies of plane curves. Our major tool will be the paper of Arnol'd [1] on the evolution of wavefronts. The sort of questions one can ask are: in a generic isotopy of a plane curve how are vertices created and destroyed? How does the dual evolve? How can the Gauss map change? In attempting to answer these questions we are taking advantage of the fact that these phenomena are all naturally associated with singularities of type Ak. Now the bifurcation set of an Ak+1 singularity and the discriminant set of an Ak singularity coincide. So we can apply Arnol'd's results on one parameter families of Legendre (discriminant) singularities (e.g. the duals) to get information on one parameter families of Lagrange (bifurcation) singularities (e.g. the evolutes). For bifurcation sets of functions with singularities other than those of type Ak one runs up against problems with smooth moduli—see [4].

scholarly journals ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES

2005 ◽  
Vol 92 (1) ◽  
pp. 99-138 ◽  
Author(s):  
J. FERNÁNDEZ DE BOBADILLA ◽  
I. LUENGO-VELASCO ◽  
A. MELLE-HERNÁNDEZ ◽  
A. NÉMETHI
Keyword(s):  
Singular Point ◽  
Projective Plane ◽  
Plane Curve ◽  
Topological Type ◽  
Homology Sphere ◽  
Plane Curves ◽  
Curve Singularity ◽  
Witten Invariant ◽  
Rational Homology ◽  
Rational Homology Sphere

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

SHADOWS, DISCRIMINANT AND DIRECT IMAGES OF PLANE CURVE SINGULARITIES

2010 ◽  
Vol 21 (04) ◽  
pp. 453-474
Author(s):  
EDUARDO CASAS-ALVERO
Keyword(s):  
Plane Curve ◽  
Plane Curves ◽  
Plane Curve Singularities ◽  
Curve Singularities

We prove that the images of irreducible germs of plane curves by a germ of analytic morphism φ have a certain contact either with branches of the discriminant of φ or with certain infinitesimal structures (shadows) that arise from the branches of the Jacobian of φ that are mapped to a point (and therefore give rise to no branch of the discriminant).

scholarly journals Freeness versus maximal global Tjurina number for plane curves

2016 ◽  
Vol 163 (1) ◽  
pp. 161-172 ◽  
Author(s):  
ALEXANDRU DIMCA
Keyword(s):  
Plane Curve ◽  
Plane Curves ◽  
Tjurina Number ◽  
Free Plane

AbstractWe give a characterisation of nearly free plane curves in terms of their global Tjurina numbers, similar to the characterisation of free curves as curves with a maximal Tjurina number, given by A. A. du Plessis and C.T.C. Wall. It is also shown that an irreducible plane curve having a 1-dimensional symmetry is nearly free. A new numerical characterisation of free curves and a simple characterisation of nearly free curves in terms of their syzygies conclude this paper.

COMPUTING MINKOWSKI SUMS OF PLANE CURVES

1995 ◽  
Vol 05 (04) ◽  
pp. 413-432 ◽  
Author(s):  
ANIL KAUL ◽  
RIDA T. FAROUKI
Keyword(s):  
Algebraic Curve ◽  
Numerical Procedure ◽  
The Other ◽  
Minkowski Sum ◽  
Plane Curves ◽  
Parametric Curves ◽  
Implicit Equation ◽  
Minkowski Sums ◽  
High Degree ◽  
The Given

The Minkowski sum of two plane curves can be regarded as the area generated by sweeping one curve along the other. The boundary of the Minkowski sum consists of translated portions of the given curves and/or portions of a more complicated curve, the “envelope” of translates of the swept curve. We show that the Minkowski-sum boundary is describable as an algebraic curve (or subset thereof) when the given curves are algebraic, and illustrate the computation of its implicit equation. However, such equations are typically of high degree and do not offer a practical basis for tracing the boundary. For the case of polynomial parametric curves, we formulate a simple numerical procedure to address the latter problem, based on constructing the Gauss maps of the given curves and using them to identifying “corresponding” curve segments that are to be summed. This yields a set of discretely-sampled arcs that constitutes a superset of the Minkowski-sum boundary, and can be regarded as a planar graph. To extract the true boundary, we present a method for identifying and “trimming” away extraneous arcs by systematically traversing this graph.

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