scholarly journals A Terracini lemma for osculating spaces with applications to Veronese surfaces

2005 ◽  
Vol 195 (1) ◽  
pp. 1-6 ◽  
Author(s):  
Edoardo Ballico ◽  
Claudio Fontanari
Keyword(s):  
Osculating Spaces

Osculating Spaces

1962 ◽  
Vol 14 ◽  
pp. 669-684
Author(s):  
Peter Scherk
Keyword(s):  
Duality Theorem ◽  
Characteristic Curve ◽  
Independent Interest ◽  
Natural Approach ◽  
Osculating Spaces ◽  
Analytical Tools

In this paper an attempt is made to prove some of the basic theorems on the osculating spaces of a curve under minimum assumptions. The natural approach seems to be the projective one. A duality yields the corresponding results for the characteristic spaces of a family of hyperplanes. A duality theorem for such a family and its characteristic curve also is proved. Finally the results are applied to osculating hyperspheres of curves in a conformai space.The analytical tools are collected in the first three sections. Some of them may be of independent interest.

scholarly journals Singular Curves of Low Degree and Multifiltrations from Osculating Spaces

2020 ◽  
Vol 2020 (21) ◽  
pp. 8139-8182 ◽  
Author(s):  
Jarosław Buczyński ◽  
Nathan Ilten ◽  
Emanuele Ventura
Keyword(s):  
Linear System ◽  
Smooth Curve ◽  
Rational Curves ◽  
Arithmetic Genus ◽  
Smooth Curves ◽  
Low Degree ◽  
Complete Linear System ◽  
Osculating Spaces ◽  
Special Case ◽  
Schubert Cycles

Abstract In order to study projections of smooth curves, we introduce multifiltrations obtained by combining flags of osculating spaces. We classify all configurations of singularities occurring for a projection of a smooth curve embedded by a complete linear system away from a projective linear space of dimension at most two. In particular, we determine all configurations of singularities of non-degenerate degree $d$ rational curves in $\mathbb{P}^n$ when $d-n\leq 3$ and $d<2n$. Along the way, we describe the Schubert cycles giving rise to these projections. We also reprove a special case of the Castelnuovo bound using these multifiltrations: under the assumption $d<2n$, the arithmetic genus of any non-degenerate degree $d$ curve in $\mathbb{P}^n$ is at most $d-n$.

Tangent Spaces of a Normal Surface with Hyperelliptic Sections

1984 ◽  
Vol 36 (1) ◽  
pp. 131-143
Author(s):  
W. L. Edge
Keyword(s):  
Normal Curve ◽  
Normal Surface ◽  
Osculating Spaces ◽  
Del Pezzo ◽  
Image Position ◽  
Rational Normal Curve

A rational normal curve C of order n in [n] has, at each point P, a nest of osculating spacesas P moves on C the [n — 2] generates a primal Dn-1 of order 2n — 2.Hilbert [3] found the multiplicities on Dn-1 not only of the vμ+1 generated by Dμ for each lesser value of μ but also those of all submanifolds common to these various Dμ.A surface Φ in higher space has, as explained [4] by del Pezzo, a nest of tangent spacesof respective dimensionsthey raise the problem of finding the orders of manifolds generated by them and the multiplicity of each on the higher manifolds to which it belongs: the task does not seem to have been attempted, but it may well be eased if Φ is rational and normal.

Topics in Direct Differential Geometry

1972 ◽  
Vol 24 (1) ◽  
pp. 98-148 ◽  
Author(s):  
Ralph Park
Keyword(s):  
Differential Geometry ◽  
Geometric Theory ◽  
Central Projection ◽  
Analytic Methods ◽  
Geometric Methods ◽  
Osculating Spaces ◽  
The Subject

In the theory of curves, one often makes differentiability assumptions in order that analytic methods can be used. Then one tries to weaken these assumptions as much as possible. The theory of curves which is presented here uses geometric methods, such as central projection, rather than analysis. In this way, no analytic assumptions are needed and a purely geometric theory results. Since this theory is not so well known as the analytic one, I have tried to make the treatment as self-contained as possible. It is hoped that this paper will form a quick introduction for a reader who has had no previous acquaintance with the subject.We assume that our curves satisfy a condition, which we call direct differentiability. Roughly this condition is that, at each point of the curve, all the osculating spaces exist.

scholarly journals The osculating spaces of a certain curve in [n]

1974 ◽  
Vol 19 (1) ◽  
pp. 39-44 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Osculating Spaces ◽  
Image Position

The curve in question is the non-singular intersection Γ of the n − 1 quadric primalswhere it is presumed that no two of the n + 1 numbers aj are equal. Definethen it will be seen that the osculating prime of Γ at x = ξ is

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