Plane curves of genus p and degree 2p as projections of space curves of the same degree

1991 ◽  
Vol 71 (1) ◽  
pp. 361-374
Author(s):  
Luisella Caire
Keyword(s):  
Plane Curves ◽  
Space Curves

scholarly journals General-affine invariants of plane curves and space curves

2019 ◽  
Vol 70 (1) ◽  
pp. 67-104
Author(s):  
Shimpei Kobayashi ◽  
Takeshi Sasaki
Keyword(s):  
Plane Curves ◽  
Space Curves ◽  
Affine Invariants ◽  
General Affine

Symmetry sets

1985 ◽  
Vol 101 (1-2) ◽  
pp. 163-186 ◽  
Author(s):  
J. W. Bruce ◽  
P. J. Giblin ◽  
C. G. Gibson
Keyword(s):  
Smooth Manifold ◽  
Normal Forms ◽  
Shape Recognition ◽  
Uniqueness Result ◽  
Potential Functions ◽  
Plane Curves ◽  
Symmetry Set ◽  
Space Curves ◽  
The Family ◽  
Set Of Points

SynopsisFor a smooth manifold M ⊆ ℝn, the symmetry set S(M) is defined to be the closure of the set of points u∈ℝn which are centres of spheres tangent to M at two or more distinct points. (The idea has its origin in the theory of shape recognition.) The connexion with singularities is that S(M) can be described alternatively as the levels bifurcation set of the family of distance-squared functions on M. In this paper a multi-germ version of the standard uniqueness result for versal unfoldings of potential functions is used to obtain a complete list of local normal forms (up to diffeomorphism) for the symmetry sets of generic plane curves, generic space curves, and generic surfaces in 3-space. For these cases the authors verify that M can be recovered as the envelope of a family of spheres centred at smooth points of S(M).

Geometry of projection-generic space curves

2009 ◽  
Vol 147 (1) ◽  
pp. 115-142
Author(s):  
C. T. C. WALL
Keyword(s):  
Local Structure ◽  
Double Point ◽  
Normal Forms ◽  
Blow Up ◽  
Complex Case ◽  
Plane Curves ◽  
Space Curves ◽  
Open Set ◽  
The Family ◽  
Flat Family

AbstractIn earlier work I defined a class of curves, forming a dense open set in the space of maps from S1 to P3, such that the family of projections of a curve in this class is stable under perturbations of C: we call the curves in the class projection-generic. The definition makes sense also in the complex case. The partition of projective space according to the singularities of the corresponding projection of C is a stratification. Its local structure outside C is the same as that of the versal unfoldings of the singularities presented.To study points on C we introduce the blow-up BC of P3 along C, and a family of plane curves, parametrised by z ∈ BC; we saw in the earlier work that this is a flat family.Here we show that near most z ∈ BC, the family gives a family of parametrised germs which versally unfolds the singularities occurring. Otherwise we find that the double point number δ of Γz drops by 1 for z ∉ EC. We establish a theory of versality for unfoldings of A or D singularities such that δ drops by at most 1, and show that in the remaining cases, we have an unfolding which is versal in this sense.This implies normal forms for the stratification of BC; further work allows us to derive local normal forms for strata of the stratification of P3.

Plane curves as projections of non singular space curves

1990 ◽  
Vol 67 (1) ◽  
pp. 433-450 ◽  
Author(s):  
Luisella Caire
Keyword(s):  
Plane Curves ◽  
Space Curves ◽  
Singular Space

scholarly journals Sur l’incomplétude de la série linéaire caractéristique d’une famille de courbes planes à nœuds et à cusps

2003 ◽  
Vol 171 ◽  
pp. 51-83 ◽  
Author(s):  
Sébastien Guffroy
Keyword(s):  
Hilbert Scheme ◽  
Singular Points ◽  
Plane Curves ◽  
Connected Space ◽  
Space Curves

AbstractSince J.Wahl ([27]), it is known that degree d plane curves having some fixed numbers of nodes and cusps as its only singularities can be represented by a scheme, let say H, which can be singular. In Wahl’s example, H is singular along a subscheme F but the induced reduced scheme Hred is smooth along F. In this work, we construct explicitly a family of plane curves with nodes and cusps which are represented by singular points of Hred.To this end, we begin to show that the Hilbert scheme of smooth and connected space curves of degree 12 and genus 15 is irreducible and generically smooth. It follows that it is singular along a hypersurface (3.10). This example is minimal in the sense that the Hilbert scheme of smooth and connected space curves is regular in codimension 1 for d < 12 (B.2). Finally we construct our plane curves from the space curves represented by points of this hypersurface (4.7).

Construction of Canal Surfaces With Variable Generatrix And Parallelism Plane Based on Plane’s Equiaffine Transformations

2017 ◽  
Vol 5 (1) ◽  
pp. 12-20 ◽  
Author(s):  
Кокарева ◽  
Ya. Kokareva
Keyword(s):  
Cross Sections ◽  
Plane Curves ◽  
Linear Variation ◽  
Space Curves ◽  
Special Cases ◽  
Canal Surfaces ◽  
Variable Step ◽  
Rotation Parameters ◽  
Surface Construction ◽  
Kinematic Approach

A kinematic approach to the construction of surfaces with equiaffine invariant for a volume of a compartment bounded by a surface, and a square of cross sections, parallel to a parallelism plane on the basis of the plane’s equiaffine transformations has been proposed in this paper. In particular, in the paper have been considered parametric equations of canal surfaces with a parallelism plane xOy, which sections have been transformed by elliptic, parabolic and hyperbolic rotations. A curve-prototype lies in the section z = 0, and a curve-image lies in the section z = h after transformation with rotations set parameters. The parametric equations have been obtained in a general form with randomness of the choice both a curve-prototype for canal surface construction, and functions defining a rate of rotation parameters change. Some general properties of the obtained surfaces have been considered, as well as some special cases of surfaces are considered, and trajectories of generating lines have been defined for them. In the case of parameter’s linear variation along the z-axis the elliptical rotation’s surfaces represent spiral and straight helical surfaces with constant and variable step however the generatrix trajectory envelops elliptical cone and cylinder. In all cases, the generatrices of the elliptical rotation’s surfaces are space curves. In the case of all parameters’ linear variation the parabolic rotation’s surfaces represent ruled surfaces. One particular case of parabolic rotation’s surfaces is the surface with two parallelism planes: xOy for equiaffine changing curve-section and xOz for generatrix. In the cases related to non-linear variation of only one parameter, generatrices for surfaces of parabolic and hyperbolic rotations are plane curves. Considered surfaces require further investigation with a view to their possible use in mechanisms and structures.

Particle and Astro-physics Challenge Kant’s Phenomenolism

1998 ◽  
pp. 323-333
Author(s):  
Lawrence H. Starkey
Keyword(s):  
General Relativity ◽  
Black Hole ◽  
Physical Basis ◽  
Primary Sources ◽  
One Dimension ◽  
Space Curves ◽  
Slowing Down ◽  
Parity Conservation ◽  
Charge Parity ◽  
Einstein’S General Relativity

For two centuries Kant's first Critique has nourished various turns against transcendent metaphysics and realism. Kant was scandalized by reason's impotence in confronting infinity (or finitude) as seen in the divisibility of particles and in spatial extension and time. Therefore, he had to regard the latter as subjective and reality as imponderable. In what follows, I review various efforts to rationalize Kant's antinomies-efforts that could only flounder before the rise of Einstein's general relativity and Hawking's blackhole cosmology. Both have undercut the entire Kantian tradition by spawning highly probable theories for suppressing infinities and actually resolving these perplexities on a purely physical basis by positing curvatures of space and even of time that make them reëntrant to themselves. Heavily documented from primary sources in physics, this paper displays time’s curvature as its slowing down near very massive bodies and even freezing in a black hole from which it can reëmerge on the far side, where a new universe can open up. I argue that space curves into a double Möbius strip until it loses one dimension in exchange for another in the twin universe. It shows how 10-dimensional GUTs and the triple Universe, time/charge/parity conservation, and strange and bottom particle families and antiparticle universes, all fit together.

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