scholarly journals On Huisman’s conjectures about unramified real curves

2021 ◽  
Vol 21 (4) ◽  
pp. 545-549
Author(s):  
Mario Kummer ◽  
Dimitri Manevich
Keyword(s):  
Normal Curve ◽  
Real Curve ◽  
Odd Degree ◽  
Real Curves ◽  
Rational Normal Curve

Abstract Let X ⊂ ℙ n be an unramified real curve with X(ℝ) ≠ 0. If n ≥ 3 is odd, Huisman [9] conjectured that X is an M-curve and that every branch of X(ℝ) is a pseudo-line. If n ≥ 4 is even, he conjectures that X is a rational normal curve or a twisted form of such a curve. Recently, a family of unramified M-curves in ℙ3 providing counterexamples to the first conjecture was constructed in [11]. In this note we construct another family of counterexamples that are not even M-curves. We remark that the second conjecture follows for generic curves of odd degree from the de Jonquières formula.

scholarly journals Equations for point configurations to lie on a rational normal curve

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

scholarly journals MINIMAL DECOMPOSITION OF BINARY FORMS WITH RESPECT TO TANGENTIAL PROJECTIONS

2013 ◽  
Vol 12 (06) ◽  
pp. 1350010 ◽  
Author(s):  
E. BALLICO ◽  
A. BERNARDI
Keyword(s):  
Normal Curve ◽  
Binary Forms ◽  
Minimal Decomposition ◽  
Tangential Projection ◽  
Rational Normal Curve

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.

The geometry and algebra of the representations of the Lorentz group

1968 ◽  
Vol 303 (1474) ◽  
pp. 381-396 ◽  
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.

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

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.

scholarly journals On extensions of Pascal's theorem

1936 ◽  
Vol 5 (1) ◽  
pp. 55-62 ◽  
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].

Projective curves of degree=codimension+2 II

2016 ◽  
Vol 26 (01) ◽  
pp. 95-104 ◽  
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].

Simplexes and other configurations upon a rational normal curve

1927 ◽  
Vol 23 (8) ◽  
pp. 882-889 ◽  
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.

scholarly journals Galois subspaces for the rational normal curve

2021 ◽  
pp. 1-10
Author(s):  
Robert Auffarth ◽  
Sebastián Rahausen
Keyword(s):  
Normal Curve ◽  
Rational Normal Curve
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