Secant Spaces to Curves

1983 ◽  
Vol 35 (4) ◽  
pp. 589-612 ◽  
Author(s):  
Joachim von Zur Gathen
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Equivalence Class ◽  
Classical Literature ◽  
Secant Variety ◽  
Rational Equivalence ◽  
Linearly Independent ◽  
Obvious Generalization ◽  
Recent Activity ◽  
Closed Subvariety

A classical question in algebraic geometry is whether a given projection of a projective space induces an isomorphism on a given closed subvariety. To answer it, one investigates secant lines to the subvariety. There has been a lot of recent activity in this field ([12], [14],[18], [21], [23]): see [14] and [12] for references).An obvious generalization of the secant lines is provided by the secant r-planes, which intersect a given closed subvariety in r + 1 linearly independent points. The closure of the set of these secant r-planes is the secant variety, and the aim of this paper is to determine its rational equivalence class in the case of curves. There is an extensive classical literature about this problem.

scholarly journals Injective linear series of algebraic curves on quadrics

2020 ◽  
Vol 66 (2) ◽  
pp. 231-254
Author(s):  
Edoardo Ballico ◽  
Emanuele Ventura
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Algebraic Curves ◽  
Linear Series ◽  
Central Importance ◽  
Arbitrary Characteristic ◽  
Space Curves

Abstract We study linear series on curves inducing injective morphisms to projective space, using zero-dimensional schemes and cohomological vanishings. Albeit projections of curves and their singularities are of central importance in algebraic geometry, basic problems still remain unsolved. In this note, we study cuspidal projections of space curves lying on irreducible quadrics (in arbitrary characteristic).

scholarly journals A wall-crossing formula for degrees of Real central projections

2014 ◽  
Vol 25 (04) ◽  
pp. 1450038 ◽  
Author(s):  
Christian Okonek ◽  
Andrei Teleman
Keyword(s):  
Algebraic Geometry ◽  
Projective Space ◽  
Pole Placement ◽  
Real Projective Space ◽  
Central Projections ◽  
Wall Crossing ◽  
Subspace Problem ◽  
Wall Crossing Formula ◽  
Real Subspace ◽  
Crucial Assumption

The main result is a wall-crossing formula for central projections defined on submanifolds of a Real projective space. Our formula gives the jump of the degree of such a projection when the center of the projection varies. The fact that the degree depends on the projection is a new phenomenon, specific to Real algebraic geometry. We illustrate this phenomenon in many interesting situations. The crucial assumption on the class of maps we consider is relative orientability, a condition which allows us to define a ℤ-valued degree map in a coherent way. We end the article with several examples, e.g. the pole placement map associated with a quotient, the Wronski map, and a new version of the Real subspace problem.

Osculatory properties of a certain curve in [n]

1974 ◽  
Vol 75 (3) ◽  
pp. 331-344 ◽  
Author(s):  
W. L. Edge
Keyword(s):  
Projective Space ◽  
Homogeneous Coordinates ◽  
Linearly Independent ◽  
Image Position

1. When, as will be presumed henceforward, no two of a0, a1, …, an are equal the n + 1 equationsare linearly independent; x0, x1, …, xn are homogeneous coordinates in [n] projective space of n dimensions—and the simplex of reference S is self-polar for all the quadrics.

scholarly journals On generalized configuration space and its homotopy groups

2020 ◽  
pp. 2043001
Author(s):  
Jun Wang ◽  
Xuezhi Zhao
Keyword(s):  
Projective Space ◽  
Configuration Space ◽  
Topological Property ◽  
Configuration Spaces ◽  
Stiefel Manifold ◽  
Homotopy Groups ◽  
Fundamental Groups ◽  
Stiefel Manifolds ◽  
Linearly Independent ◽  
Special Cases

Let [Formula: see text] be a subset of vector space or projective space. The authors define generalized configuration space of [Formula: see text] which is formed by [Formula: see text]-tuples of elements of [Formula: see text], where any [Formula: see text] elements of each [Formula: see text]-tuple are linearly independent. The generalized configuration space gives a generalization of Fadell’s classical configuration space, and Stiefel manifold. Denote generalized configuration space of [Formula: see text] by [Formula: see text]. For studying topological property of the generalized configuration spaces, the authors calculate homotopy groups for some special cases. This paper gives the fundamental groups of generalized configuration spaces of [Formula: see text] for some special cases, and the connections between the homotopy groups of generalized configuration spaces of [Formula: see text] and the homotopy groups of Stiefel manifolds. It is also proved that the higher homotopy groups of generalized configuration spaces [Formula: see text] and [Formula: see text] are isomorphic.

scholarly journals CALABI QUASIMORPHISMS FOR THE SYMPLECTIC BALL

2004 ◽  
Vol 06 (05) ◽  
pp. 793-802 ◽  
Author(s):  
PAUL BIRAN ◽  
MICHAEL ENTOV ◽  
LEONID POLTEROVICH
Keyword(s):  
Projective Space ◽  
Open Subset ◽  
Complex Projective Space ◽  
Clifford Torus ◽  
Compactly Supported ◽  
Linearly Independent ◽  
Calabi Invariant ◽  
Hofer Metric ◽  
Complex Projective

We prove that the group of compactly supported symplectomorphisms of the standard symplectic ball admits a continuum of linearly independent real-valued homogeneous quasimorphisms. In addition these quasimorphisms are Lipschitz in the Hofer metric and have the following property: the value of each such quasimorphism on any symplectomorphism supported in any "sufficiently small" open subset of the ball equals the Calabi invariant of the symplectomorphism. By a "sufficiently small" open subset we mean that it can be displaced from itself by a symplectomorphism of the ball. As a byproduct we show that the (Lagrangian) Clifford torus in the complex projective space cannot be displaced from itself by a Hamiltonian isotopy.

Genera of rationally equivalent integral binary quadratic forms

1991 ◽  
Vol 119 (1-2) ◽  
pp. 27-30 ◽  
Author(s):  
A. G. Earnest
Keyword(s):  
Equivalence Class ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Binary Quadratic Forms ◽  
Rational Equivalence ◽  
Necessary And Sufficient

SynopsisA formula is given for the number of genera of primitive integral binary quadratic forms of discriminant D which lie in a rational equivalence class. In particular, necessary and sufficient conditions for the genus and rational equivalence class to coincide are given in terms of the prime factorisation of D.

scholarly journals The degree of the tangent and secant variety to a projective surface

2020 ◽  
Vol 20 (2) ◽  
pp. 233-248
Author(s):  
Andrea Cattaneo
Keyword(s):  
Projective Space ◽  
Hilbert Scheme ◽  
Projective Surface ◽  
Secant Variety ◽  
Smooth Projective Surface

AbstractWe present a way of computing the degree of the secant (resp. tangent) variety of a smooth projective surface, under the assumption that the divisor giving the embedding in the projective space is 3-very ample. This method exploits the link between these varieties and the Hilbert scheme 0-dimensional subschemes of length 2 of the surface.

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