MINIMAL DECOMPOSITION OF BINARY FORMS WITH RESPECT TO TANGENTIAL PROJECTIONS

Let C ⊂ ℙn+1 be a rational normal curve and let X ⊂ ℙn be one of its tangential projection. We describe the X-rank of a point P ∈ ℙn in terms of the schemes evincing the C-rank or the border C-rank of the preimage of P.

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The geometry of the binary quintic form

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100018144 ◽

1944 ◽

Vol 40 (1) ◽

pp. 1-5 ◽

Cited By ~ 2

Author(s):

J. A. Todd

Keyword(s):

Complete System ◽

Normal Curve ◽

Binary Forms ◽

Five Dimensions ◽

Rational Normal Curve

1. It is familiar that the properties of the invariants and covariants of binary forms of the first four orders admit of elegant geometrical interpretations on rational normal curves and their projections. For forms of order higher than four the number of irreducible concomitants which appear in the complete system increases rapidly. It is the purpose of this note to exhibit geometrically the 23 irreducible concomitants of the binary quintic, using the rational normal curve R5 in space of five dimensions and its projection R′5 on to a prime as a foundation.

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Equations for point configurations to lie on a rational normal curve

Advances in Mathematics ◽

10.1016/j.aim.2018.10.013 ◽

2018 ◽

Vol 340 ◽

pp. 653-683

Author(s):

Alessio Caminata ◽

Noah Giansiracusa ◽

Han-Bom Moon ◽

Luca Schaffler

Keyword(s):

Normal Curve ◽

Point Configurations ◽

Rational Normal Curve

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The geometry and algebra of the representations of the Lorentz group

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1968.0056 ◽

1968 ◽

Vol 303 (1474) ◽

pp. 381-396 ◽

Cited By ~ 7

Keyword(s):

Projective Space ◽

Lorentz Group ◽

Point Of View ◽

Normal Curve ◽

Geometrical Representation ◽

Spinor Space ◽

Dimensional Projective Space ◽

And Algebra ◽

Rational Normal Curve ◽

Formal Point

In this paper a (2j + l)-spinor analysis is developed along the lines of the 2-spinor and 3-spinor ones. We define generalized connecting quantities A μv (j) which transform like (j, 0) ⊗ (j -1, 0) in spinor space and like second rank tensors under transformations in space-time. The general properties of the A uv are investigated together with algebraic relations involving the Lorentz group generators, J μv . The connexion with 3j symbols is discussed. From a purely formal point of view we introduce a geometrical representation of a (2j +1)-spinor as a point in a 2j dimensional projective space. Then, for example, the charge conjugate of a (2j + l)-spinor is just the polar of the corresponding point with respect to a certain rational, normal curve in the projective space. It is suggested that this representation will prove useful.

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A Specialised Net of Quadrics Having Selfpolar Polyhedra, with Details of the Fivedimensional Example

Canadian Journal of Mathematics ◽

10.4153/cjm-1981-069-3 ◽

1981 ◽

Vol 33 (4) ◽

pp. 885-892

Author(s):

W. L. Edge

Keyword(s):

Projective Space ◽

Complex Number ◽

Roots Of Unity ◽

Parametric Form ◽

Normal Curve ◽

Homogeneous Coordinates ◽

Image Position ◽

Rational Normal Curve

If x0,x1, … xn are homogeneous coordinates in [n], projective space of n dimensions, the prime (to use the standard name for a hyperplane)osculates, as θ varies, the rational normal curve C whose parametric form is [2, p. 347]Take a set of n + 2 points on C for which θ = ηjζ where ζ is any complex number andso that the ηj, for 0 ≦ j < n + 2, are the (n + 2)th roots of unity. The n + 2 primes osculating C at these points bound an (n + 2)-hedron H which varies with η, and H is polar for all the quadrics(1.1)in the sense that the polar of any vertex, common to n of its n + 2 bounding primes, contains the opposite [n + 2] common to the residual pair.

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On extensions of Pascal's theorem

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500008282 ◽

1936 ◽

Vol 5 (1) ◽

pp. 55-62 ◽

Cited By ~ 1

Author(s):

H. W. Richmond

Keyword(s):

Striking Feature ◽

Normal Curve ◽

Rational Normal Curve ◽

Closed Polygon

The object of this paper is firstly to extend the theorem of Pascal concerning six points of a conic to sets of 2 (n + 1) points of the rational normal curve of order n in space of n dimensions; secondly to explain why a wider extension to other sets of 2 (n + 1) points in [n] must be sought; and lastly to give briefly an extension to [3] and [4] which will be further generalised in a later paper. The striking feature of Pascal's theorem—that each of the sixty ways of arranging the points in a cycle, or as vertices of a closed polygon, leads to a different version of the theorem—is retained in the following extension to [n].

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Projective curves of degree=codimension+2 II

International Journal of Algebra and Computation ◽

10.1142/s0218196716500041 ◽

2016 ◽

Vol 26 (01) ◽

pp. 95-104 ◽

Cited By ~ 1

Author(s):

Wanseok Lee ◽

Euisung Park

Keyword(s):

Integral Curve ◽

Betti Numbers ◽

Free Resolution ◽

Relative Location ◽

Normal Curve ◽

Minimal Free Resolution ◽

Graded Betti Numbers ◽

Projective Curves ◽

Rational Normal Curve

Let [Formula: see text] be a nondegenerate projective integral curve of degree [Formula: see text] which is not linearly normal. In this paper, we continues the study begun in [E. Park, Projective curves of degree=codimension+2, Math. Z. 256 (2007) 685–697] for the minimal free resolution of [Formula: see text]. It is well-known that [Formula: see text] is an isomorphic projection of a rational normal curve [Formula: see text] from a point [Formula: see text]. Our main result is about how the graded Betti numbers of [Formula: see text] are determined by the rank of [Formula: see text] with respect to [Formula: see text], which is a measure of the relative location of [Formula: see text] from [Formula: see text].

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Simplexes and other configurations upon a rational normal curve

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100013748 ◽

1927 ◽

Vol 23 (8) ◽

pp. 882-889 ◽

Cited By ~ 2

Author(s):

F. P. White

Keyword(s):

Normal Curve ◽

Common Tangent ◽

Rational Normal Curve

The theorem that if two triangles be inscribed in a conic their six sides touch another conic is, of course, to be found in all the text-books; it is apparently due in the first place to Brianchon. The further remark, that if three triangles be inscribed in a conic the three conics obtained from them in pairs have a common tangent, is to be found in Taylor's Ancient and Modern Geometry of Conics; it was made independently by Wakeford.

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